FULL-SCALE MEASUREMENTS AND SYSTEM IDENTIFICATION:
A TIME-FREQUENCY PERSPECTIVE
VOLUME I
A Dissertation
Submitted to the Graduate School
of the University of Notre Dame
in Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
by
Tracy Lynn Kijewski-Correa, B.S.C.E., M.S.C.E.
_________________________________
Ahsan Kareem, Director
Department of Civil Engineering and Geological Sciences
Notre Dame, Indiana
April 2003
FULL-SCALE MEASUREMENTS AND SYSTEM IDENTIFICATION:
A TIME-FREQUENCY PERSPECTIVE
Abstract
by
Tracy Lynn Kijewski-Correa
The rapid development of the global, urban landscape propels structures to new heights
and spans, enhancing the role of structural engineers in assuring a safe and habitable built
environment. In this new era of high-rise buildings and long-span bridges, design
challenges continually arise, motivating the need to better understand the dynamic
behavior of these structures through full-scale monitoring. In the context of such
programs, the role of inherent damping, which proves critical for this flexible class of
structures, must be quantified with accuracy – a challenging endeavor considering that
narrowbanded response must be analyzed often without the benefit of measured input.
More significantly, due to the presence of nonlinear and nonstationary characteristics in
the measured data, traditional analysis tools may be incapable of capturing some of the
salient features. This dissertation addresses these very issues by discussing the estimation
of inherent damping from ambient data and evaluating its uncertainty via bootstrap
resampling; by introducing a comprehensive wavelet-based analysis framework tailored
to accommodate nonlinear and nonstationary features of Civil Engineering signals; and
finally by introducing an on-going tall building monitoring program in Chicago, which
Tracy Lynn Kijewski-Correa
includes the configuration, calibration and application of Global Positioning Systems for
the dynamic monitoring of displacements.
In particular, the wavelet framework introduced in this research, predicated on the
properties of the Morlet Wavelet, provides flexible criteria for the separation of closelyspaced modal contributions, discretization of the time-frequency plane, and identification
and melioration of end effects through a reflective padding scheme. Within this
framework, wavelets are demonstrated to be capable of revealing hidden characteristics
in simulated and measured wind, wave and earthquake data. In these discussions, the
wavelet instantaneous bandwidth is introduced as a means to track subcyclic nonlinear
behavior in a manner comparable to the Hilbert Spectra. Other wavelet applications in
this study include its adaptation for identification of correlation in time and frequency
and its enhancement through smart thresholding schemes. This framework also provides
the backdrop for the extension of wavelets to the identification of frequency and damping
in Civil Engineering structures, for which a number of processing concerns become
paramount.
For My Family,
who have shown me that
with love, faith and family
nothing is impossible…
ii
CONTENTS
VOLUME I
TABLES ..............................................................................................................................x
FIGURES......................................................................................................................... xiii
ACKNOWLEDGEMENTS........................................................................................... xxix
CHAPTER 1: OVERVIEW.................................................................................................1
1.1 Civil Engineering’s Enigma.....................................................................................2
1.2 Moving into a New Analysis Domain .....................................................................3
1.3 Call to Full-Scale .....................................................................................................5
CHAPTER 2: Damping Identification and Associated Uncertainty....................................8
2.1 Introduction..............................................................................................................8
2.2 Traditional Approaches to Damping Identification from Wind-Induced
Response ................................................................................................................11
2.2.1 Spectral Analysis and Inherent Errors .......................................................12
2.2.2 Random Decrement Technique..................................................................17
2.2.2.1 Types of Triggering Conditions .......................................................22
2.2.2.2 Investigation of Factors Influencing RDT Quality ..........................24
2.2.2.2.1 Suggested Expression for Random Decrement
Signature Variance ..........................................................26
2.2.2.2.2 Significance of Overlap Between Captured
Segments..........................................................................28
2.2.2.2.3 Required Number of Captured Segments..........................29
2.2.2.2.4 Significance of Triggering Condition on Damping
Estimation........................................................................30
2.2.2.2.5 Influence of Nonstationarity on Random
Decrement Signatures......................................................45
2.3 Uncertainty in System Identification and Damping Estimation ............................48
2.4 Statistical Analysis of Damping Estimates via Bootstrapping ..............................54
iii
2.4.1
Discussion: Bootstrapping Damping Estimates for Spectral
Analysis......................................................................................................59
2.4.2 Discussion: Bootstrapping Damping Estimates for Random
Decrement Technique ................................................................................62
2.5 The Challenges of Nonstationarity and Nonlinearity ............................................66
2.6 Conclusions............................................................................................................70
2.7 Moving into the Time-Frequency Domain ............................................................71
CHAPTER 3: TRANSITION TO THE TIME-FREQUENCY DOMAIN .......................73
3.1 Departure from Fourier Analysis ...........................................................................73
3.2 Short-Time Fourier Transform ..............................................................................75
3.2.1 Heisenberg Uncertainty Principle ..............................................................76
3.2.2 Resolutions of the STFT: The Heisenberg Box.........................................78
3.3 Moving Toward the Wavelet Domain ...................................................................79
3.4 The Transform .......................................................................................................81
3.4.1 Completeness .............................................................................................82
3.4.2 Conditions on Parent Wavelets..................................................................84
3.5 Wavelet Resolutions: The Heisenberg Box ...........................................................85
3.6 Discretization and Orthogonality...........................................................................87
3.7 Wavelet Transforms and Analytic Signals ............................................................89
3.7.1 Instantaneous Frequency............................................................................89
3.7.2 Theory of Asymptotic Signals ...................................................................92
3.7.3 Theory of Wavelet Ridges .........................................................................93
3.7.4 Asymptotic Signals and Mechanical Oscillators .......................................94
3.7.5 Alternate Methods to Generate Analytic Signals for MultiComponent Data ........................................................................................95
3.8 Benefits of the Wavelet Transform for Civil Engineering Applications ...............98
CHAPTER 4: FRAMEWORK FOR WAVELET-BASED ANALYSIS .......................100
4.1 Necessity for Unified Analysis Framework.........................................................100
4.2 Morlet Wavelets...................................................................................................101
4.2.1 Frequency Resolution: Tuning the Central Frequency ............................103
4.2.2 Temporal Resolution: A Physical Understanding of End Effects ...........107
4.2.2.1 End Effects: Influence of End Effects on Spectral Amplitude.......111
4.2.2.2 End Effects: Influence on Spectral Bandwidth Estimation............114
4.2.2.3 End Effects: Melioration via Reflective Padding...........................118
4.2.2.4 End Effects: Efficacy of Reflective Padding Scheme ....................122
4.2.2.5 End Effects: Comparison of Zero Padding and Reflective
Padding...........................................................................................128
4.2.2.6 Tuning Temporal Separation..........................................................129
4.3 Discretization of the Time-Frequency Plane .......................................................130
4.4 Ridge Extraction Techniques...............................................................................135
4.5 Conclusions..........................................................................................................140
iv
CHAPTER 5: WAVELET SYSTEM IDENTIFICATION I: APPLICATIONS TO
WIND, WAVE AND EARTHQUAKE ANALYSIS .....................................................142
5.1 Introduction..........................................................................................................142
5.2 Preliminaries ........................................................................................................143
5.3 Example ...............................................................................................................151
5.4 Earthquakes..........................................................................................................158
5.4.1 El Centro (1940) ......................................................................................160
5.4.2 Mexico City (1985)..................................................................................175
5.4.3 Loma Prieta (1989) ..................................................................................189
5.4.4 Northridge (1994) ....................................................................................199
5.4.5 Kobe (1995) .............................................................................................208
5.4.6 Comparison of Earthquake Properties .....................................................217
5.4.7 Seismic Response of a Building ..............................................................221
5.5 Wind-Induced Response of a Tall Building.........................................................227
5.5.1 Preliminaries ............................................................................................228
5.5.2 Global Analysis........................................................................................232
5.5.3 Event 1: 0-200 s .......................................................................................236
5.5.4 Event 2: 1350-1550 s ...............................................................................246
5.5.5 Event 4: 2600-2800 s ...............................................................................247
5.5.6 Event 5: 3000-3200 s ...............................................................................249
5.5.7 Conclusions Based on Wavelet Spectral Analysis...................................252
5.6 Offshore Platforms...............................................................................................254
5.7 Summary ..............................................................................................................268
CHAPTER 6: WAVELET SYSTEM IDENTIFICATION II: APPLICATIONS
TO WAVELET COHERENCE.......................................................................................269
6.1 Introduction..........................................................................................................269
6.2 Wavelet Coherence Background: Scalogram and Coscalogram .........................271
6.3 Wavelet-based Coherence Map ...........................................................................273
6.3.1 Comparison of Wavelet- and Fourier-based Coherence Estimates .........275
6.3.2 Application of Wavelet-Coherence to Non-stationary Signals................277
6.4 Minimization of Spurious Coherence ..................................................................279
6.4.1 Multi-Resolution Integration Windows ...................................................281
6.4.2 Ridge Extraction by Hard Thresholding ..................................................288
6.4.3 Filtered Wavelet Coherence Map Based on White Noise .......................291
6.4.4 Filtered Wavelet Coherence Map Based on Signal Characteristics.........293
6.5 Scale/Frequency-averaged Wavelet Coherence Map ..........................................299
6.6 Conclusions..........................................................................................................299
CHAPTER 7: WAVELET SYSTEM IDENTIFICATION III: ANALOGS TO
HILBERT SPECTRAL ANALYSIS...............................................................................303
7.1 Introduction..........................................................................................................303
7.2 Theory of Hilbert Spectral Analysis ....................................................................304
v
7.3 Preliminaries ........................................................................................................307
7.3.1 Resolutions of Hilbert Spectral Analysis.................................................307
7.3.2 End Effects in Hilbert Spectral Analysis .................................................308
7.3.3 Introduction of Instantaneous Bandwidth................................................308
7.4 Proper Conditions for Comparison of Hilbert Spectral Analysis and
Wavelet Transforms.............................................................................................313
7.5 Examples..............................................................................................................315
7.5.1 Example 1: Localized Sine Wave ............................................................315
7.5.2 Example 2: Sine Wave with Frequency Discontinuity ............................317
7.5.3 Example 3: Quadratic Chirp ....................................................................317
7.5.4 Example 4: Stokes Wave .........................................................................320
7.5.5 Example 5: Linear Sum of Two Closely-Spaced Cosines .......................327
7.5.6 Example 6: Amplitude Modulated Signal with Constant Frequency ......330
7.5.7 Example 7: Duffing Equation ..................................................................339
7.5.8 Example 8: Lorenz Equation ...................................................................347
7.5.9 Example 9: Rössler Equation...................................................................352
7.5.10 Example 10: Random Wave Data............................................................357
7.6 Discussion: Processing Concerns and Performance in Noise..............................360
7.7 Discussion: A Fundamental Difference ...............................................................364
7.8 Conclusions..........................................................................................................366
CHAPTER 8: WAVELET SYSTEM IDENTIFICATION IV: FREQUENCY
AND DAMPING ESTIMATION FOR CIVIL ENGINEERING STRUCTURES.........368
8.1 Introduction..........................................................................................................368
8.2 Previous Applications of Wavelets for System Identification.............................369
8.3 Influence of End Effects in System Identification: Spectral Measures ...............372
8.4 Influence of End Effects in System Identification: Time Domain ......................374
8.5 System Identification for Free Vibration: MDOF Example ................................376
8.5.1 Ridge Extraction and Wavelet Instantaneous Spectra .............................379
8.5.2 Wavelet Skeletons and End Effects .........................................................382
8.5.3 System Identification via Wavelet Amplitude and Phase........................383
8.6 System Identification from Ambient Vibration: MDOF Example ......................389
8.7 System Identification from Ambient Vibration Data: Full-Scale Example.........395
8.8 Conclusions..........................................................................................................402
VOLUME II
TABLES .................................................................................................................... xxxviii
FIGURES........................................................................................................................... xl
CHAPTER 9: OVERVIEW OF FULL-SCALE MONITORING PROGRAM .............404
9.1 Introduction..........................................................................................................404
vi
9.2 The Need for Full-Scale Monitoring of Tall Buildings .......................................405
9.2.1 Enhancing the Understanding of In-Situ Damping..................................406
9.2.2 Broader Impacts of Monitoring Program.................................................408
9.3 Recent Full-Scale Monitoring Programs .............................................................409
9.4 Monitored Structures ...........................................................................................412
9.4.1 Building 1.................................................................................................413
9.4.2 Building 2.................................................................................................413
9.4.3 Building 3.................................................................................................414
9.5 Primary Instrumentation System..........................................................................414
9.5.1 Accelerometers ........................................................................................415
9.5.2 Data Loggers and Supporting Electronics ...............................................419
9.5.3 Installation and Placement .......................................................................422
9.6 Anemometers .......................................................................................................424
9.7 Data Transmission and Management...................................................................428
9.8 Example of Full-Scale Data .................................................................................430
9.9 Summary ..............................................................................................................433
CHAPTER 10: INTRODUCTION TO GLOBAL POSITIONING SYSTEMS ............434
10.1 Introduction........................................................................................................434
10.2 Origins of GPS Satellite Network......................................................................434
10.3 The GPS Concept...............................................................................................435
10.4 Anatomy of GPS Satellite Signals .....................................................................438
10.5 Safeguards..........................................................................................................442
10.6 Inherent GPS Errors and Corrective Configurations .........................................442
10.6.1 Differential GPS (DGPS).........................................................................443
10.6.2 Differential Phase Positioning .................................................................444
10.7 Residual GPS Errors ..........................................................................................447
10.7.1 Dilution of Precision (DOP) Errors .........................................................448
10.7.2 Multi-Path Errors .....................................................................................451
10.8 Motivation for Structural Monitoring Applications...........................................453
10.8.1 Recent Applications to Civil Engineering Structures ..............................455
10.8.2 Insights from Previous Research .............................................................456
10.9 Summary ............................................................................................................459
CHAPTER 11: OVERVIEW OF GLOBAL POSITIONING SYSTEM AND
CALIBRATION TESTS..................................................................................................460
11.1 Introduction........................................................................................................460
11.2 GPS Components ...............................................................................................460
11.2.1 GPS Receiver ...........................................................................................461
11.2.2 GPS Antenna............................................................................................462
11.2.3 Accuracy ..................................................................................................463
11.2.4 GPS Hardware Configuration ..................................................................464
11.2.5 Lightning Protection System....................................................................467
11.2.6 GPS Data Acquisition Software...............................................................469
11.3 Field Site for Experimental Validation ..............................................................470
vii
11.4 Test Configuration .............................................................................................470
11.5 Overview of Tests ..............................................................................................474
11.5.1 Test 1a-c: Verification of Background Noise and Influence of
Satellite Position ......................................................................................475
11.5.2 Test 2a-w: Verification of Amplitude Sensitivity....................................476
11.5.3 Test 3a-f: Verification of Ability to Track Complex Signals ..................477
11.5.4 Test 4a-c: Influence of Gas Capsule ........................................................479
11.5.5 Test 5: Coordinate Transformation Mock-Up .........................................480
11.5.6 Test 6a-b: Influence of Antenna Mount...................................................482
11.6 Post-Processing Protocol ...................................................................................483
11.6.1 Cutoff Angle ............................................................................................484
11.6.2 Baseline Limit..........................................................................................485
11.6.3 RMS Threshold........................................................................................485
11.6.4 Solution Type...........................................................................................486
11.6.5 Ionospheric Modeling ..............................................................................486
11.6.6 Stochastic Modeling.................................................................................488
11.6.7 Tropospheric Modeling............................................................................488
11.6.8 Single Point Processing............................................................................489
11.7 Interpretation of Results.....................................................................................490
11.8 Summary ........................................................................................................... 491
CHAPTER 12: GLOBAL POSITIONING SYSTEM CALIBRATION TEST
RESULTS AND DISCUSSION ......................................................................................492
12.1 Introduction........................................................................................................492
12.2 Verification of Background Noise and Influence of Satellite Position:
Results..................................................................................................................492
12.2.1 Probability and Spectral Structure of Background Noise ........................496
12.2.2 Additional Information from Static Component of Dynamic Tests ........500
12.3 Accuracy Estimates............................................................................................502
12.4 Data Preparation.................................................................................................508
12.5 Analysis of Dynamic Calibration Tests .............................................................510
12.5.1 Test 2a-w: Verification of Amplitude Sensitivity....................................513
12.5.2 Test 3a-3f: Verification of Ability to Track Complex Signals ................523
12.6 Test 4a-c: Influence of Gas Capsule ..................................................................531
12.7 Test 5: Coordinate Transformation Mock-Up ...................................................539
12.8 Test 6a-b: Influence of Antenna Mount.............................................................541
12.9 Recommendations for Full-Scale Application...................................................545
CHAPTER 13: GPS MONITORING IN URBAN ENVIRONMENTS: DATA
MINING AND INFORMATION PROCESSING...........................................................548
13.1 Introduction........................................................................................................548
13.2 Identification of Reference Site .........................................................................548
13.3 Installation of GPS Components in Full-Scale ..................................................549
13.3.1 Antenna Mounts.......................................................................................549
13.3.2 Grounding ................................................................................................552
viii
13.3.3 Receiver Cabinetry...................................................................................553
13.4 Monitoring Program...........................................................................................554
13.4.1 Modifications to the Post–Processing Routine ........................................557
13.4.2 Preliminary Baseline Position..................................................................558
13.5 Radio Frequency Interference............................................................................562
13.6 Multi-Path Interference and Identification.........................................................563
13.7 Assessment of System Performance in Full-Scale.............................................574
CHAPTER 14: CONCLUSIONS AND FUTURE DIRECTIONS .................................576
14.1 Contributions of This Work ...............................................................................576
14.1.1 Wavelet Analysis Framework and Application to Civil
Engineering Signals .................................................................................577
14.1.2 Wavelet Analogs to Hilbert Spectral Analysis ........................................578
14.1.3 Introduction of Full-Scale Monitoring and Advanced
Instrumentation: Global Positioning Systems..........................................579
14.1.4 Evaluation and Treatment of Uncertainty in Damping Estimation
from Ambient Vibration Data..................................................................580
14.2 Future Directions ...............................................................................................581
14.2.1 Extension of Framework to Other Wavelets............................................581
14.2.2 Enhancing Ridge Extraction Abilities .....................................................582
14.2.3 Window Separation..................................................................................583
14.2.4 Enhancement of GPS Sensing Technologies...........................................584
APPENDIX: OVERVIEW OF RESAMPLING THEORY: THE BOOTSTRAP ..........586
A.1 Motivation...........................................................................................................586
A.2 Bootstrap Theory.................................................................................................587
A.3 Bootstrapping to Estimate Variance in RDS and PSD .......................................588
WORKS CITED ............................................................................................................. 593
ix
TABLES
VOLUME I
2.1
Standard deviations of simulated time histories ....................................................33
2.2
Estimated LogDec damping values using positive point triggers..........................36
2.3
Repeatability and local averaging for Run I, Local extrema trigger,
ΤRDS = 0..................................................................................................................38
2.4
Estimated damping values using local extrema trigger .........................................39
2.5
Influence of absolute value trigger conditions.......................................................42
2.6
Properties of simulated cases .................................................................................51
2.7
Statistics of Monte Carlo simulations ....................................................................52
2.8
Statistics of bootstrap replications of critical damping ratio: estimates by
spectral analysis .....................................................................................................60
2.9
Statistics of bootstrap replications of critical damping ratio: estimates by
RDT........................................................................................................................64
5.1
Influence of overlap factor on wavelet energy estimates.....................................152
5.2
Relative contributions of each component to instantaneous spectra for
example ................................................................................................................158
5.3
Resolutions of Fourier and wavelet analyses.......................................................161
5.4
Signal standard deviation as estimated from Fourier and wavelet marginal
spectra ..................................................................................................................164
5.5
Relative contributions of each component to instantaneous spectra for El
Centro earthquake primary and secondary analyses............................................171
5.6
Relative contributions of each component to instantaneous spectra for
Guerrero MC Nearfield earthquake primary and secondary analyses .................185
x
5.7
Relative contributions of each component to instantaneous spectra for
Loma Prieta Earthquake primary and secondary analyses ..................................198
5.8
Relative contributions of each component to instantaneous spectra for
Northridge earthquake primary and secondary analyses .....................................208
5.9
Relative contributions of each component to instantaneous spectra for
Kobe earthquake primary and secondary analyses ..............................................216
5.10
Comparison of energetic components and energy accumulation levels in
time and frequency for five earthquake records ..................................................218
5.11
Relative contributions of each component to instantaneous spectra for
Sherman Oaks building in Northridge earthquake...............................................226
8.1
Estimate of damping from wavelet skeleton of SDOF system ............................375
8.2
Wavelet-based identification of MDOF system with closely spaced modes.......379
8.3
Wavelet frequency and damping estimates from RDS (Track A) and IRF
(Track B)..............................................................................................................391
8.4
Wavelet-based identification from full-scale ambiently-excited data .................401
VOLUME II
9.1
Summary of accelerometer properties considered...............................................417
9.2
Specifications of ultrasonic anemometer with heater ..........................................427
9.3
Specifications of interim wind sensor..................................................................428
10.1
Multiplication factors to generate coherent signals from P-code ........................441
10.2
Summary of inherent errors in GPS and proposed solutions...............................454
11.1
Accuracy levels for Leica RTK GPS ...................................................................464
11.2
Environmental constraints on GPS equipment ....................................................466
11.3
Overview of calibration tests ...............................................................................478
12.1
Satellite conditions and DOP errors for static tests .............................................493
12.2
Statistics of GPS static displacements .................................................................495
xi
12.3
Comparison of 99.7th percentile confidence limits of GPS background
noise with those of Gaussian distribution ............................................................499
12.4
Statistics of E-W (static) direction in dynamic tests............................................501
12.5
Standard deviation of GPS static displacements compared to average of
standard deviation in GPS displacement estimate ...............................................504
12.6
Chebyshev filter cut-off frequencies....................................................................510
12.7
Statistics of σN and mean noise threshold...........................................................512
12.8
Accuracy of GPS tracking for Test 2...................................................................514
12.9
Accuracy of GPS tracking for Test 3...................................................................524
12.10 Statistics of GPS displacement estimates and associated errors for Test
4a-c.......................................................................................................................535
12.11 Estimates of motion along shifted axis for Test 5................................................541
12.12 Statistics of GPS displacement estimates and associated errors for Test 6 .........543
13.1
Monitored dates of interest ..................................................................................558
13.2
Preliminary baseline position determination .......................................................561
xii
FIGURES
VOLUME I
2.1
Schematic of half-power bandwidth method of spectral system
identification. .........................................................................................................16
2.2
Conceptualization of the Random Decrement Technique, for local
extrema triggering condition..................................................................................19
2.3
(top to bottom) RDT against theoretical autocorrelation, bootstrap
variance envelope, bootstrapped variance estimate against Equation
2.17, theoretical variance in Equation 2.13, uncorrelated segments......................27
2.4
(top to bottom) RDT against theoretical autocorrelation, bootstrap
variance envelope, bootstrapped variance estimate against Equation
2.12, theoretical variance in Equation 2.13, correlated segments..........................29
2.5
RDT cyclic variance (top) and resulting damping estimation error. .....................31
2.6
Time histories of SDOF response for Runs I-IV ...................................................33
2.7
Histograms (left) and power spectra for simulated response in Runs IIV ...........................................................................................................................34
2.8
Captured random decrement segments for Run III, Xo=11.9. Dotted line
indicates theoretical free vibration decay at this amplitude level ..........................45
2.9
Bootstrap variance estimates for non-stationary signal 1 with error
expression in Equation 2.22 shown by solid line (Nr=200) ...................................47
2.10
Bootstrap variance estimates for nonstationary signal 2........................................48
2.11
Identified dynamic properties of simulated data: identification of Case
B by (a) spectral analysis and (b) Random Decrement Technique........................52
2.12
Histogram of bootstrap simulations on (a) PSD and (b) RDS for Case B.............58
2.13
Amplitude-dependent damping model...................................................................68
xiii
3.1
Sine wave with local frequency change (top) and traditional power
spectral magnitude .................................................................................................74
3.2
Concept of Heisenberg box for short-time Fourier Transforms ............................79
3.3
Sum of two hyperbolic chirps and resulting spectrogram (Mallat, 1998) .............80
3.4
Evolution of time-frequency analysis ....................................................................81
3.5
Wavelet transform of two hyperbolic chirps shown in Figure 3.3
(Mallat, 1998) ........................................................................................................82
3.6
Concept of Heisenberg box for Wavelet Transforms ............................................87
3.7
Schematic representation of asymptotic signal conditions ....................................93
4.1
Comparison of basis functions for Fourier and Morlet Wavelet
Transforms ...........................................................................................................102
4.2
Real and imaginary components of Morlet wavelet for various central
frequencies ...........................................................................................................104
4.3
Effective frequency and time resolutions of Morlet wavelet and
associated Heisenberg box...................................................................................106
4.4
Schematic demonstration the implication of parameter selection for
modal separation ..................................................................................................107
4.5
Illustration of region susceptible to end effects ...................................................110
4.6
Real component of Morlet wavelet with Gaussian window emphasized ............110
4.7
Scalogram of sine wave along ridge. Dotted line denotes theory and
solid line is calculated result. Vertical bars demarcate end effects
regions, β∆ti, for β=1-5........................................................................................112
4.8
Deviations of simulated instantaneous spectra (gray) from theoretical
result (black) as end-effects regions are progressively neglected........................113
4.9
Schematic representation of half-power bandwidth estimation...........................115
4.10
Improvements made in half-power bandwidth estimates by successively
neglecting larger end effects regions. Theoretical prediction (dashed)
and calculated result.............................................................................................117
4.11
Even and odd signals and their respective reflections (dotted line).....................120
4.12
Concept of signal padding for arbitrary function.................................................120
xiv
4.13
Simulated time series for validation of padding scheme .....................................123
4.14
Superposition of instantaneous spectra over all time for calculated
Wavelet Transform (gray) with theoretical prediction (black) ............................125
4.15
Superposition of instantaneous spectra over all time for calculated
Wavelet Transform (gray) with theoretical prediction (black) with
padding operation.................................................................................................126
4.16
Efficiacy of padding operation to reduce end effects in wavelet
bandwidth measures, theoretical prediction (dashed) and simulation
(solid): (a) first mode half-power bandwidth without padding, (b) fifth
mode half-power bandwidth without padding, (c) tenth mode halfpower bandwidth without padding, (d) first mode half-power
bandwidth with padding and β = 6, (e) fifth mode half-power
bandwidth with padding and β = 6, (f) tenth mode half-power
bandwidth with padding and β = 6 ......................................................................127
4.17
(a) 2 Hz cosine; (b) 2 Hz sine – influence of padding on wavelet
skeleton magnitude ..............................................................................................129
4.18
Implications of overlapping analyzing frequencies .............................................135
4.19
Schematic of various time-frequency characteristics of Wavelet
Transforms ...........................................................................................................137
5.1
Spectra of environmental loads acting on Civil Engineering structures
(taken from Kareem, 1987)..................................................................................143
5.2
Admissibility function for 2 Hz Morlet wavelet..................................................146
5.3
Example signal, wavelet scalogram and WIFS (left, top to bottom) and
marginal spectral comparison with Fourier power spectrum (right). ..................153
5.4
Energy accumulation in frequency domain (left) and time domain. ...................155
5.5
Signal and rate of change of energy accumulation in time domain (left)
and marginal spectrum and rate of change of energy accumulation in
frequency domain for example cosine signal.......................................................156
5.6
Wavelet instantaneous spectra taken at critical time steps in the
example signal evolution t=20 s, 48.5 s, 51 s, 75 s..............................................157
5.7
El Centro ground motion, wavelet scalogram and WIFS (top to bottom)
for primary (left) and secondary analyses............................................................161
5.8
Wavelet marginal spectra with Fourier power spectrum for primary
(left) and secondary analyses for El Centro quake. .............................................164
xv
5.9
Energy accumulation in frequency domain and time domain for
primary (left) and secondary analysis of El Centro. ............................................165
5.10
Wavelet marginal spectrum and rate of change of energy accumulation
in frequency domain for primary (left) and secondary analysis of El
Centro...................................................................................................................165
5.11
El Centro ground acceleration and rate of change of energy
accumulation in time domain for primary (left) and secondary analysis.............166
5.12
Wavelet instantaneous spectra taken at critical time steps in the El
Centro event for primary (top) and secondary analysis. ......................................169
5.13
(a) Fourier spectrum and (b) low frequency zoom; (c) Hilbert marginal
spectrum and (d) low frequency zoom for El Centro analysis conducted
by Huang et al. (1998). ........................................................................................173
5.14
Ten IMF components from El Centro data provided by Huang et al.
(1998)...................................................................................................................174
5.15
Guerrero MC nearfield ground motion, wavelet scalogram and WIFS
(top to bottom) for primary (left) and secondary analyses. .................................177
5.16
Wavelet marginal spectra with Fourier power spectrum for primary
(left) and secondary analyses for Guerrero MC nearfield record. .......................178
5.17
Energy accumulation in frequency domain and time domain for
primary (left) and secondary analysis of Guerrero MC nearfield record.............179
5.18
Wavelet marginal spectrum and rate of change of energy accumulation
in frequency domain for primary (left) and secondary analysis of
Guerrero MC nearfield motion. ...........................................................................179
5.19
Guerrero MC nearfield ground acceleration and rate of change of
energy accumulation in time domain for primary (left) and secondary
analysis.................................................................................................................180
5.20
Wavelet instantaneous spectra taken at critical time steps in the
Guerrero MC nearfield event for primary (top) and secondary analysis. ............183
5.21
Comparison of Fourier spectrum and wavelet marginal spectrum (left)
for Mexico City farfield ground motion with wavelet scalogram and
WIFS (right, top to bottom). ................................................................................187
5.22
Energy accumulation in frequency and time (left), accompanied by
Mexico City nearfield ground acceleration and rate of change of energy
accumulation in time domain...............................................................................188
xvi
5.23
Wavelet instantaneous spectra taken at critical time steps in the Mexico
City farfield record...............................................................................................189
5.24
Loma Prieta ground motion, wavelet scalogram and WIFS (top to
bottom) for primary (left) and secondary analyses. .............................................191
5.25
Wavelet marginal spectra with Fourier power spectrum for primary
(left) and secondary analyses for Loma Prieta quake. .........................................192
5.26
Energy accumulation in frequency domain and time domain for
primary (left) and secondary analysis of Loma Prieta. ........................................193
5.27
Wavelet marginal spectrum and rate of change of energy accumulation
in frequency domain for primary (left) and secondary analysis of Loma
Prieta. ...................................................................................................................194
5.28
Loma Prieta ground acceleration and rate of change of energy
accumulation in time domain for primary (left) and secondary analysis.............195
5.29
Wavelet instantaneous spectra taken at critical time steps in the Loma
Prieta event for primary (top) and secondary analysis.........................................196
5.30
Northridge ground motion, wavelet scalogram and WIFS (top to
bottom) for primary (left) and secondary analyses. .............................................200
5.31
Wavelet marginal spectra with Fourier power spectrum for primary
(left) and secondary analyses for Northridge quake. ...........................................202
5.32
Energy accumulation in frequency domain and time domain for
primary (left) and secondary analysis of Northridge. ..........................................202
5.33
Wavelet marginal spectrum and rate of change of energy accumulation
in frequency domain for primary (left) and secondary analysis of
Northridge. ...........................................................................................................203
5.34
Northridge ground motion and rate of change of energy accumulation
in time domain for primary (left) and secondary analysis. ..................................203
5.35
Wavelet instantaneous spectra taken at critical time steps in the
Northridge event for primary (top) and secondary analysis. ...............................206
5.36
Kobe ground motion, wavelet scalogram and WIFS (top to bottom) for
primary (left) and secondary analyses. ................................................................210
5.37
Wavelet marginal spectra with Fourier power spectrum for primary
(left) and secondary analyses for Kobe quake. ....................................................210
xvii
5.38
Energy accumulation in frequency domain and time domain for
primary (left) and secondary analysis of Kobe. ...................................................212
5.39
Wavelet marginal spectrum and rate of change of energy accumulation
in frequency domain for primary (left) and secondary analysis of Kobe. ...........212
5.40
Kobe ground acceleration and rate of change of energy accumulation in
time domain for primary (left) and secondary analysis. ......................................213
5.41
Wavelet instantaneous spectra taken at critical time steps in the Kobe
event for primary (top) and secondary analysis...................................................215
5.42
Comparison of energy accumulation curves in frequency domain for
each of the five nearfield earthquake records. .....................................................218
5.43
Comparison of energy accumulation curves in frequency domain for
each of the five nearfield earthquake records. .....................................................220
5.44
(a) Recorded structural response to Northridge ground motion, (b)
Fourier and wavelet marginal spectra; (c) scalogram; (d) WIFS; (e)
zoom of third mode ridge; (f) zoom of second mode ridge; (g) zoom of
fundamental mode ridge. .....................................................................................222
5.45
Energy accumulation in frequency domain and time domain (left) and
structural accelerations and rate of change of energy accumulation in
time domain for Sherman Oaks building under Northridge earthquake..............224
5.46
Wavelet instantaneous spectra taken at critical time steps in the
Sherman Oaks building response in the Northridge earthquake..........................226
5.47
Response components for buildings under the action of wind loading
(taken from Kijewski et al., 2001) .......................................................................228
5.48
Plan view of monitored building with accelerometer locations and
orientations...........................................................................................................229
5.49
(a) Comparison of wavelet marginal spectrum (solid) and Fourier
spectrum (dotted) for beating sines; (b) beating sine signal and wavelet
instantaneous spectra at 6 locations in the time history.......................................231
5.50
Wavelet marginal spectra for the four sensors at the 57th floor of
Boston building....................................................................................................233
5.51
Time histories of acceleration response detected at four accelerometer
locations. Events analyzed herein denoted by arrows. ........................................235
5.52
Local wavelet marginal spectrum for #59-EW (Event 1)....................................237
xviii
5.53
Wind speed, direction and response at #59-EW (Event 1) with wavelet
instantaneous spectral analysis. ...........................................................................238
5.54
Response at #02-EW (Event 1) with wavelet instantaneous spectral
analysis.................................................................................................................241
5.55
Response at #59-NS (Event 1) with wavelet instantaneous spectral
analysis.................................................................................................................243
5.56
Response at #02-NS (Event 1) with wavelet instantaneous spectral
analysis.................................................................................................................245
5.57
Response at #59-NS (Event 2) with wavelet instantaneous spectral
analysis.................................................................................................................247
5.58
Response at #59-EW (top) and #02-EW (Event 4) with wavelet
instantaneous spectral analysis ............................................................................248
5.59
WIFS for #02-NS (Event 4).................................................................................250
5.60
Response at #02-EW (Event 5) with wavelet instantaneous spectral
analysis.................................................................................................................251
5.61
WIFS for #02-EW (Event 5)................................................................................252
5.62
Schematic of response components for tension leg platforms (taken
from Kareem, 1985).............................................................................................255
5.63
Fourier power spectra for wave field in (a) mild sea state and (b) severe
sea state ................................................................................................................257
5.64
Severe sea state (a) surge response, (b) scalogram, (c) WIFS and
instantaneous spectra at tj = (d) 1007.8 s, (e) 1542.2 s, (f) 1970.3 s, (g)
2155.0 s ................................................................................................................257
5.65
Comparison of surge response in mild (blue) and severe (red) sea states
(a) wavelet marginal spectrum (solid) and Fourier spectrum (dotted);
(b) energy accumulation with frequency .............................................................260
5.66
Mild sea state (a) surge response, (b) scalogram, (c) WIFS and
instantaneous spectra at tj = (d) 891.6 s, (e) 1168.0 s, (f) 1459.1 s, (g)
1772.2 s ................................................................................................................261
5.67
Comparison of sway response in mild (blue) and severe (red) sea states
(a) wavelet marginal spectrum (solid) and Fourier spectrum (dotted);
(b) energy accumulation with frequency .............................................................263
5.68
Mild sea state (a) sway response, (b) scalogram and (c) WIFS...........................264
xix
5.69
Severe sea state (a) sway response, (b) scalogram ..............................................265
5.70
Comparison of yaw-induced lateral response in mild (blue) and severe
(red) sea states (a) wavelet marginal spectrum (solid) and Fourier
spectrum (dotted); (b) energy accumulation with frequency...............................266
5.71
(a) Yaw-induced lateral response for mild sea state and (b) scalogram;
(c) yaw-induced sway response for severe sea state and (d) scalogram ..............267
6.1
(a) Scalogram of upstream wind velocity 1; (b) scalogram of rooftop
pressure; (c) coscalogam of these two correlated processes; (d)
scalogram of upstream wind velocity 2; (e) scalogram of rooftop
pressure; (f) coscalogram of these two uncorrelated processes (taken
from Gurley & Kareem, 1999).............................................................................273
6.2
(a) Incoming wave surface elevation; (b) TLP surge response; (c)
wavelet and FFT coherence estimates between wave elevation and TLP
response; (d) wavelet and FFT coherence estimates between wave
elevation and TLP response with incoherent nose added to each........................276
6.3
Coherence between v2(t) and pr(t) and between v1(t) and pr(t) ...........................278
6.4
Wavelet coherence map between v1(t) and pr(t), also shown as semilogarithmic in frequency, emphasizing low frequency content ...........................280
6.5
Influence of central frequency on the occurrence of spurious coherence
in wavelet coherence map. Known coherent pockets between v1(t) and
pr(t) designated by boxes.....................................................................................282
6.6
Illustration of variable integration window concept ............................................284
6.7
Examples of variance reduction by variable integration window........................287
6.8
Examples of coarse ridge extraction by thresholding ..........................................290
6.9
Filtered wavelet coherence map between v1(t) and pr(t) with g selected
for varying noise exceedence levels, based on white noise .................................294
6.10
Filtered wavelet coherence map between v1(t) and pr(t) with g selected
for varying noise exceedence levels, based on signal characteristics..................296
6.11
(a) Measured wind velocity; (b) measured wind pressure; (c) unfiltered
wavelet coherence map; filtered wavelet coherence map with g selected
for varying levels of noise exceedence: (d) 25% exceedence; (e) 10%
exceedence; (f) 1% exceedence ...........................................................................298
xx
6.12
(a) Scale-averaged coherence along with mean coherence and noise
threshold; (b) Hermite polynomial-based PDF model of reference noise
maps .....................................................................................................................300
7.1
Example 1: (a) isolated cycle of 1 Hz sine wave; (b) wavelet
scalogram; (c) WIFS (red dotted) with frequency of sine wave (blue
dashed); (d) Huang et al.’s (1998) Morlet wavelet result; (e) Huang et
al.’s (1998) Hilbert spectrum ...............................................................................316
7.2
Example 2: (a) cosine wave with frequency halved midway through
signal; (b) wavelet scalogram; (c) WIFS; (d) Huang et al.’s (1998)
Hilbert spectrum; (e) Huang et al.’s (1998) Morlet wavelet result......................318
7.3
Example 3: (a) quadratic chirp; (b) wavelet scalogram; (c) identified
instantaneous frequency by WT (in blue) and HT (in red); (d) wavelet
scalogram using padded signal and finer discretization of scales; (e)
identified instantaneous frequency by WT (in blue) and HT (in red) and
actual quadratic frequency law (green) using padded signal and finer
discretization of scales .........................................................................................319
7.4
Example 4: (a) Second order approximation to the Stokes wave; (b)
wavelet modulus at t=250 s; (c) wavelet scalogram; (d) WIFS; (e)
Huang et al.’s (1998) Morlet wavelet result; (f) Huang et al.’s (1998)
Hilbert spectrum...................................................................................................322
7.5
Example 4 via refined wavelet analysis: (a) second order approximation
to the Stokes wave; (b) wavelet scalogram; (c) WIFS; (d) wavelet
instantaneous half-power bandwidth ...................................................................324
7.6
(a) 30 seconds wave tank surface elevation; (b) scalogram of wave tank
data; (c) WIFS; (d) instantaneous frequency from wavelet phase; (e)
instantaneous half-power bandwidth estimate; (f) instantaneous
frequency from Hilbert transform phase..............................................................325
7.7
Example 5: (a) linear combination of two closely-spaced cosine waves;
(b) wavelet scalogram; (c) wavelet scalogram (zoom); (d) WIFS; (e)
Huang et al.’s (1998) Morlet wavelet analysis with Hilbert spectrum
contours superimposed.........................................................................................328
7.8
Example 6: (a) amplitude-modulated cosine; (b) wavelet scalogram; (c)
WIFS; (d) wavelet scalogram with application of padding; (e) WIFS
with application of padding (zoom); (f) WIFS determined from wavelet
phase with application of padding (zoom); (g) wavelet estimate of
HPBW with application of padding; (h) 3D view of wavelet scalogram
with application of padding .................................................................................331
7.9
(a) Huang et al.’s (1998) Morlet wavelet result; (b) Huang et al.’s
(1998) Hilbert spectrum (zoom) ..........................................................................332
xxi
7.10
(a) Amplitude-modulated cosine in Example 6; (b) Fourier spectrum of
envelope a(t); (c) Fourier spectrum of phase cos(φ(t)); (d) Amplitudemodulated cosine in with envelope and phase well separated in
frequency; (e) Fourier spectrum of envelope a(t); (f) Fourier spectrum
of phase cos(φ(t)) .................................................................................................336
7.11
Instantaneous frequencies identified from the Hilbert transform: (a)
case where envelope and phase spectra overlap significantly; (b) case
where envelope and phase spectra overlap somewhat (Ex. 6), shown in
Figure 7.8; (c) case where envelope and phase spectra do not overlap
significantly..........................................................................................................337
7.12
Example 7: (a) Duffing oscillator under harmonic excitation; (b) Huang
et al.’s (1998) Hilbert spectrum manifesting 3 IMF components; (c)
Huang et al.’s (1998) Morlet wavelet result ........................................................340
7.13
Example 7: (a) Duffing oscillator in free vibration, zoomed in from 50
to 100 s; (b) wavelet scalogram; (c) WIFS; (d) wavelet instantaneous
frequency spectra at each time step plotted one atop another; (e)
instantaneous frequency identified by phase of WT; (f) instantaneous
half-power bandwidth ..........................................................................................341
7.14
Example 7: (a) Duffing oscillator in forced vibration; (b) wavelet
scalogram; (c) WIFS; (d) wavelet instantaneous frequency spectra at
each time step plotted one atop another; (e) instantaneous frequency of
high frequency component identified by phase of WT; (f) instantaneous
half-power bandwidth of high frequency component; (g) estimate of
instantaneous frequency by Hilbert Transform....................................................344
7.15
Example 8: (a) Lorenz equation x-component; (b) Huang et al.’s (1998)
Hilbert spectrum of Lorenz equation, x-compnent; (c) Huang et al.’s
(1998) Morlet wavelet result................................................................................348
7.16
Example 8: (a) Lorenz equation x-component; (b) wavelet scalogram;
(c) WIFS; (d) instantaneous frequency of high frequency component
identified by phase of WT; (e) instantaneous half-power bandwidth of
high frequency component...................................................................................350
7.17
Example 8: Lorenz equation, x-component, instantaneous spectrum
from wavelet transform at critical points in the response ....................................352
7.18
Example 9: (a) Rössler equation x-component; (b) Huang et al.’s
(1998) Hilbert spectrum; (c) Huang et al.’s (1998) Morlet wavelet
result.....................................................................................................................354
7.19
Example 9: (a) Rössler equation x-component; (b) WIFS for analysis
stage 1; (c) sample instantaneous spectra; (d) wavelet scalogram for
xxii
analysis stage 2; (e) WIFS for analysis stage 2; (f) instantaneous halfpower bandwidth for analysis stage 2..................................................................355
7.20
Example 10: (a) measured random wave data; (b) wavelet scalogram;
(c) WIFS for up to three ridges (red → green indicates highest to lowest
energy components at each time step) .................................................................358
7.21
Example 10: instantaneous frequency identified from (a) single mode
spectra; (b) bi-modal spectra; (c) multi-mode spectra (red → green
indicates highest to lowest energy components at that time step), with
example of each instantaneous spectra class shown at the right..........................359
7.22
(a) Real component of Morlet wavelet basis function for fo=5/(2π) Hz;
(b) real component of Morlet wavelet basis function for fo=5 Hz .......................362
8.1
(a) Signal; (b) & (e) signal (dark) and wavelet skeleton (light) at ends
without padding; (c) & (f) signal (dark) and wavelet skeleton (light) at
ends with padding; (d) & (g) signal (dark) and difference between
signal and wavelet skeleton (light) at ends with padding; (e) bandwidth
estimate with & without padding.........................................................................373
8.2
Impulse response function of MDOF system with closely spaced modes...........377
8.3
Wavelet system identification framework, Track A for ambient
vibrations or Track B for measured free vibration or impulse response..............380
8.4
Instantaneous frequencies identified from ridges of wavelet transform
for (a) fo = 3 Hz, (b) fo = 6 Hz, (c) fo = 8 Hz and (d) fo = 6 Hz with
padding.................................................................................................................381
8.5
Instantaneous wavelet spectra for (a) fo = 3 Hz when two modes are
present, (b) fo = 3 Hz when three modes are present, (c) fo = 6 Hz and
(d) fo = 8 Hz .........................................................................................................382
8.6
Real component of wavelet skeleton – each panel contains skeleton for
the ridge associated with modes 1, 2 and 3, respsectivley, for (a) fo = 3
Hz, (b) fo = 6 Hz, (c) fo = 8 Hz and (d) fo = 6 Hz with padding. Dotted
vertical line demarks the 3∆ti region of end effects.............................................384
8.7
Wavelet phase and amplitude curves for system identification of
MDOF system (solid) with linear least squares fit (dotted), for (a) fo = 6
Hz and (b) fo = 8 Hz. Each panel contains data for modes 1, 2 and 3
from left to right...................................................................................................384
8.8
Identification of damping from piecewise fit to amplitude curves of
MDOF system for (a) fo = 6 Hz, (b) fo = 6 Hz with padding, (c) fo = 8
Hz, and (d) fo = 8 Hz with padding. Each panel contains data for modes
xxiii
1, 2 and 3 from left to right. Dotted vertical line demarks the 3∆ti, 4∆ti
and 5∆ti end effects regions .................................................................................386
8.9
Identification of damping via logarithmic decrement of wavelet
skeleton for fo = 8 Hz with padding; modes 1, 2 and 3 shown from left
to right..................................................................................................................388
8.10
Random decrement signatures (solid) in comparison to actual impulse
response function (dashed) for 2DOF system......................................................390
8.11
Frequency and damping identification by RDT and wavelet transform
framework for repeated simulations of (a) Case 1 and (b) Case 2.......................392
8.12
Wavelet-based system identification from (a) random decrement
signatures and (b) impulse response functions ....................................................394
8.13
Example of random decrement signature (gray) and variance envelope.............396
8.14
Ten-minute segment of the acceleration response (y-dir) of tower in
typhoon ................................................................................................................397
8.15
(a) Random decrement signature, (b) real component of wavelettransformed Random Decrement Signature in 3D; (c) real-valued
skeleton; (d) zoom of real-valued skeleton (solid) with theoretical
skeleton (dotted) for fn = 0.625 Hz and ξ = 0.015 ...............................................398
8.16
(a) Frequency and (c) damping identified from wavelet skeleton, zoom
of (b) frequency and (d) damping estimates, (e) wavelet phase along
ridge, (f) wavelet amplitude along ridge..............................................................400
VOLUME II
9.1
Primary instrumentation system for full-scale monitoring program....................415
9.2
Operational range of Wilcoxon sensor (courtesy of Wilcoxon) ..........................418
9.3
Columbia accelerometer (left) and accelerometer pair in enclosure
ready for mounting...............................................................................................420
9.4
Data logger cabinet assembly installed in Building 1 (zoomed in for
detail) and comparable logger assembly for Building 2 (inset at right) ..............421
9.5
Positions of accelerometers (1, 2) and data loggers (3) in Building 1
and Building 2......................................................................................................423
9.6
Mounted accelerometer housing in Building 1 (left) and Building 2 ..................423
xxiv
9.7
Photo of Viasala ultrasonic anemometer .............................................................426
9.8
Accelerometer data from Building 1 on 11/30/02 from 1:00 to 2:00: (a)
alongwind component; (b) acrosswind component; (c) torsion-induced
alongwind component at corner; (d) torsion-induced acrosswind
component at corner.............................................................................................431
9.9
Fourier power spectra of accelerometer data from Building 1 on
11/30/02 from 1:00 to 2:00: (a) alongwind component; (b) acrosswind
component; (c) torsion-induced alongwind component at corner; (d)
torsion-induced acrosswind component at corner................................................432
10.1
GPS strategy for determining position.................................................................437
10.2
Relation of position in WGS84 coordinate system to latitude and
longitude on Earth’s surface (taken from Leica, 1999) .......................................437
10.3
Schematic representation of GPS satellite signal structure..................................440
10.4
Schematic of elevation and azimuth measures for a GPS satellite at
135°azimuth and 60° elevation ............................................................................449
10.5
Schematic representation of satellite configurations leading to low DOP
(left) and high DOP (adapted from Leica, 1999).................................................450
10.6
Errors in differential phase data manifesting long-period multi-path
errors (taken from Axelrad et al., 1996) ..............................................................453
11.1
Choke ring antenna (top, left), outfitted with protective dome covering
(top, right) and GPS receiver (bottom) ................................................................462
11.2
Configuration of GPS data acquisition system installed in each building...........466
11.3
Huber + Suhner lighting protector with gas capsule............................................468
11.4
Demonstration of 15° mask angle requirement limiting neighboring
obstructions ..........................................................................................................471
11.5
GPS reference and rover antennas affixed to rigid mounts .................................472
11.6
Orientation of reference and rover station for Tests 1-4......................................473
11.7
Views in each direction of test site ......................................................................474
11.8
Predicted availability of satellites and dilution of precision for
Anderson Road site on January 22, 2002 (screen capture from Leica
software) ..............................................................................................................476
xxv
11.9
Predicted availability of satellites and dilution of precision for
Anderson Road site on April 17, 2002 (screen capture from Leica
software) ..............................................................................................................480
11.10 Photo of gas capsule assembly I in Test 4a..........................................................481
11.11 Orientation of reference and rover stations for Tests 5-6 ....................................483
11.12 Sample SKI-Pro output (screen capture from Leica software)............................491
12.1
Portion of time history of GPS relative displacement for static tests
Test 1b and Test 1c ..............................................................................................495
12.2
Results from static tests and comparisons between observed RMS
displacement (inner box) and manufacturer’s prediction (outer box) .................497
12.3
Probability density (compared to Gaussian function with same mean
and standard deviation) with vertical bars denoting 1, 2 and 3 standard
deviations of the mean, and power spectral density for each of the static
tests in Test 1 series, E-W component .................................................................498
12.4
Probability density (compared to Gaussian function with same mean
and standard deviation) with vertical bars denoting 1, 2 and 3 standard
deviations of the mean, and power spectral density for each of the static
tests in Test 1 series, N-S component ..................................................................499
12.5
Standard deviation of background noise along E-W direction, (static)
for all tests, as a function of GDOP .....................................................................502
12.6
Spectral structure of position quality measure for Test 1a and Test 2v,
along with probability distribution for Test 1a ....................................................506
12.7
Average position quality as a function of GDOP ................................................507
12.8
Examples of noise threshold levels (dotted) [cm] superimposed on GPS
displacements [cm] for a series of static and dynamic tests ................................508
12.9
(a) Test 2a GPS displacement data; (b) Test 2a GPS displacement data
after low-pass filtering; (c) Test 2e GPS displacement data; (d) Test 2e
GPS displacement data after low-pass filtering ...................................................509
12.10 (a) PSD of Test 2a GPS displacement data; (b) PSD of Test 2a GPS
displacement data after low-pass filtering; (c) PSD of Test 2e GPS
displacement data; (d) PSD of Test 2e GPS displacement data after
low-pass filtering .................................................................................................511
12.11 Comparison of shake table displacement (red) to GPS displacement
estimate for Tests 2a-2f, mean noise threshold shaded in red .............................516
xxvi
12.12 Comparison of shake table displacement (red) to GPS displacement
estimate for Tests 2g-2l, mean noise threshold shaded in red .............................518
12.13 Comparison of shake table displacement (red) to GPS displacement
estimate for Tests 2m-2r, mean noise threshold shaded in red ............................520
12.14 Comparison of shake table displacement (red) to GPS displacement
estimate for Tests 2s-2w, mean noise threshold shaded in red ............................522
12.15 Comparison of shake table displacement (red) to GPS displacement
estimate for Test 3a at various intervals in the test, mean noise
threshold shaded in red ........................................................................................526
12.16 Comparison of shake table displacement (red) to GPS displacement
estimate for Test 3b at various intervals in the test, mean noise
threshold shaded in red ........................................................................................527
12.17 Comparison of shake table displacement (red) to GPS displacement
estimate for Test 3c over the duration of the test (top) and zooming in
at various intervals in the test, mean noise threshold shaded in blue ..................528
12.18 Comparison of shake table displacement (red) to GPS displacement
estimate for Test 3d over the duration of the test (top) and zooming in
at various intervals in the test, mean noise threshold shaded in blue ..................530
12.19 Comparison of shake table displacement (red) to GPS displacement
estimate for Test 3e over the duration of the test (top) and zooming in
at various intervals in the test, mean noise threshold shaded in blue ..................532
12.20 Comparison of shake table displacement (red) to GPS displacement
estimate for Test 3f over the duration of the test (top) and zooming in at
various intervals in the test, mean noise threshold shaded in blue ......................533
12.21 GPS displacements estimated during Tests 4a-c (left) and standard
deviation of GPS displacement estimate..............................................................536
12.22a Time histories of Test 4a-c GPS East-West displacement predictions
and standard deviation of GPS displacement estimate ........................................537
12.22b Time histories of Test 4a-c GPS North-South displacement predictions
and standard deviation of GPS displacement estimate ........................................538
12.23 Generalized orientation of reference and rover stations for Test 5......................540
12.24 Comparison of actual table displacement (red) and transformed GPS
displacement estimates along E-W and N-S axes, using transformation
angle of 45° and 50°............................................................................................542
xxvii
12.25 GPS displacement estimates for Test 6a-b and standard deviations of
GPS displacement estimate..................................................................................544
13.1
Reference and rover antenna mounts fabricated for full-scale
application............................................................................................................551
13.2
Schematic of GPS antenna placement on rooftop frame of Building 1...............552
13.3
Fully installed GPS antennas at Building 1/rover site (left) and
reference site ........................................................................................................553
13.4
In-line lightning protection with grounding wire (left) and installed in
full-scale at reference site ....................................................................................554
13.5
Zoom of GPS cabinetry contents installed in full-scale program ........................555
13.6
GPS Instrumentation cabinet in place at reference site (left) and at rover
site just below data logger cabinet in Building 1.................................................556
13.7
Preliminary baseline position from monitoring on 11/14/2002 and
comparison with displaced position on 11/13/2002.............................................561
13.8
Fourier power spectra of relative displacements to the (a) north and (b)
east: original data in black, filtered data in gray..................................................566
13.9
Filtered displacement data for 11/30/02 ..............................................................567
13.10 Resonant displacement data for 11/30/02 with error thresholds (green) .............568
13.11 Zoom of resonant displacement data for 11/30/02 ..............................................569
13.12 East and north relative displacements on consecutive sidereal days
(data for 1/08/03 has been shifted 4 minutes)......................................................571
13.13 Quasi-static east and north relative displacements on consecutive
sidereal days (data for 1/08/03 has been shifted 4 minutes) ................................572
13.14 Resonant east and north relative displacements on consecutive sidereal
days (data for 1/08/03 has been shifted 4 minutes)..............................................573
A.1
Schematic diagram of generalized bootstrap concept (adapted from
Efron & Tibshirani, 1993)....................................................................................588
A.2
Bootstrapping scheme for system identification..................................................590
A.3
Variance envelopes for (a) RDT and (b) PSD. Grey lines indicate
variance envelope; black line indicates traditional RDT and PSD
estimate; dotted lines indicate RDS decay and HPBW .......................................591
xxviii
ACKNOWLEDGEMENTS
This dissertation serves as a testament to the quality of a Notre Dame education and all it
has inspired in me. I am especially fortunate to have been given the opportunity to attend
this university for each of my degrees, during which I have been surrounded by some of
my dearest friends and most respected colleagues. As much as the faculty of Notre Dame
has been instrumental in my training and fostering, they are but a part of a larger Catholic
community — the “Notre Dame family” — that has supported me during the most
difficult times of my life. This family includes the many dedicated faculty who have
served as my educators, my committee of Drs. Kareem, Chang, Kirkner and Kurama who
took great time and care to read and critique this research, and the department’s support
staff over the years: Tammy, Judy, Mollie, Chris, Carrie and Denise, who kept me on
track and kept this whole operation running. However, I realize that none of this would
have been possible without the vision of my advisor and friend, Ahsan Kareem, who saw
the potential of a junior in his Structural Analysis course and opened the world of
research and academia to her. Thank you, “bossman,” for never losing faith in me and in
what I could become one day. Only two men have affected me in such a profound way,
and I hope you know that you are one of them. Thank you for challenging me, mentoring
me, inspiring me and ultimately guiding me toward a future that is brighter than any I
could have imagined.
xxix
During my time at Notre Dame, I had opportunities I would never have found
anywhere else — none more apparent than the full-scale monitoring program introduced
in this work. I am grateful first and foremost to the engineers, owners and managers of
these buildings for their continued support and cooperation with this research program. I
would also like to thank the management of the Old Republic Building in Chicago for
serving as the reference station for our GPS monitoring program. Particularly, in the
development of this GPS framework, I must recognize a number of individuals: my
mother and her staff for coordinating the installation and maintenance of the GPS
reference station, John Berkebile of the St. Joseph County Parks Department for allowing
us access to the Anderson Road site, Derrik Miles and James Stowell of Leica
Geosystems for their advice in configuring the sensor, Brent Bach for helping to
assemble the mounts and cabinetry, and Dr. Kwon, Tiphaine and Lijuan for helping to
calibrate the system and post-process the GPS data. However, in the larger scope of my
research, I have collaborated with a number of individuals and organizations who have
greatly enriched my work, and therefore I would also like to thank at this time: my
colleagues at Skidmore Owings and Merrill in Chicago and the Boundary Layer Wind
Tunnel Laboratory at the University of Western Ontario, Dr. Yukio Tamura of the Tokyo
Institute of Polytechnics, Mr. Frank Durgin formerly of MIT, Dr. Yin Zhou of Malouf
Engineering, and Dr. Kurtis Gurley of the University of Florida at Gainesville.
Over the last six years at Notre Dame, I have been surrounded by so many
amazing friends that I am truly privileged to know: Tim, Laura, Dan, my roommates over
the years: Gale, Kate, Rachel, Lisa, Joy and Katy, Enos, the Dessanyakes, Shelley, Greg,
the Sims, the Van Dykes, the Ohtoris, Bill, Nathan, Pete, the Morgens, Nelson, Andrea,
xxx
Ethan, Katie, Clara, Tony, Dana, Naru, Erica, Sara, Brad, Kelly, George, my office
mates: Kara, Karrie-Ann and Ann, and everyone else that has been a part of our
CE/GEOS family. Thank you all for helping me to create Notre Dame moments of
laughter and love. In particular, I would like to thank Amy and Kenny for blessing me
with the sister and brother I never had. And I would be remiss if I forgot to recognize my
other family at Notre Dame — “Team Kareem” — for providing intellectual stimulus,
support and ultimately friendship: Kurt, Mike, Fred, Luigi, Gang, Swaroop, Dr. Ohdo,
Dr. Kanda, Tiphaine, Devin, Lijuan, Dr. Chen, Dr. Zhou and Dr. Kwon.
Of course, none of these experiences and relationships would have ever
materialized had I not come to Notre Dame, and I attribute that great achievement to my
family, beginning with my mother who gave me life and had the courage and strength to
raise me on her own. Thank you, Mommy, for showing everyone that we could manage
just fine. But thankfully, we were not truly alone, and my life was so much richer for the
grace, love and guidance of my late grandparents and my aunts and uncles. Over the
years, our family has grown with the addition of my cousins, my stepfather and my inlaws, who have continued to provide me with a supportive, nurturing environment
grounded deeply in faith and family that empowered us all throughout the stressful years
of my graduate education and my grandparents’ illnesses. My one pillar through all these
trials and challenges has been my husband Marco, who always was my biggest
cheerleader, my strongest ally, my deepest confidant, and my best friend. Marco, I am so
fortunate to have you in my life. I also cannot forget my little fuzzy friend who relaxes
me every night after a long day at work: Bob the cat.
xxxi
Finally, I would like to gratefully acknowledge support from National Science
Foundation Grant CMS 00-85109, the National Science Foundation Graduate Research
Trainee program, the National Defense Science and Engineering Graduate Fellowship
through the Department of Defense, the NASA Indiana Space Grant, the Center for
Applied Mathematics at the University of Notre Dame and the additional resources and
support of the Department of Civil Engineering and Geological Sciences.
Note: PDF versions of this dissertation are available, in color, at the following
website: www.nd.edu/~nathaz.
xxxii
CHAPTER 1
OVERVIEW
The rapid development of the global, urban landscape continues to propel structures to
new heights and spans, enhancing the role of structural engineers in assuring a safe and
habitable built environment. In this new era of high-rise buildings and long-span bridges,
the advancements in the state-of-the-art have presented new design challenges and
unveiled dynamic behaviors that were not previously considered, but prove critical for
this new breed of structure. Just as the limits of engineering design are tested in these
ambitious projects, so too must the limits of the research which supports them, in the
form of interactive design tools to better quantify the response of tall buildings in both
sway and torsion (Zhou et al., 2003), in full-scale monitoring projects to better
understand the in-situ dynamic response of these structures in real wind environments,
and in innovative time-frequency analysis tools to investigate nonlinear and nonstationary
characteristics in both their response and environmental loading. The following study
primarily explores these latter two areas to demonstrate that the new challenges facing
Civil Engineering can only be addressed by pushing the envelope, and in doing so, laying
the foundation for the next generation of advancements.
1
1.1 Civil Engineering’s Enigma
The challenges that face Civil Engineering practice begin and end with the basic equation
of motion:
MX&& + CX& + KX = F
(1.1)
where M is the mass, K is the stiffness and C is the damping coefficient. X is the
displacement response of the structure to some external force F, where the dots above the
displacement variable are indicative of the derivative order. A great concern of Civil
Engineering comes from the right side of this equation, in the search for accurate
quantification of the forces or demand on the structure, often complicated by the
stochastic nature of naturally occurring loads such as earthquakes, waves and wind.
Meanwhile, the evolving complexity of the left hand side of this equation, corresponding
to structural capacity, will continue to challenge Civil Engineers, in the face of
nonlinearities, modal coupling and self-excited forces arising from modern and futuristic
structures. However, before even moving on to more complicated versions of this
expression and absent any external load, there is still great uncertainty in even this most
basic equation. That uncertainty lies in damping, a true enigma in structural dynamics
(Gurley & Kareem, 1996; Kijewski & Kareem, 2000). The second chapter of this study
will explore the sources of uncertainty surrounding structural damping, the challenges
associated with its estimation, and the viability of existing techniques for its extraction
from full-scale, ambient vibration data, invoking a bootstrap resampling scheme detailed
in the Appendix of this study to investigate the variance of damping estimates.
2
1.2 Moving into a New Analysis Domain
The discussions of structural damping and its estimation challenges in Chapter 2
highlight an important observation: many physical processes of interest to Civil
Engineers manifest nonlinear and nonstationary features. As a result, their complete
characterization may not be accomplished via Fourier Transforms, necessitating a new
analysis framework. Chapter 3 chronicles the evolution from the Fourier domain to a
time-frequency domain home to the multi-resolution Wavelet Transform. As motivated in
Chapter 3, the dual nature of wavelet transforms, being a simultaneous transform in both
time and frequency, justifies its recent extension to the analysis of stochastic processes of
interest to Civil Engineering, adapting the transform to a number of situations where
Fourier transforms were traditionally used to define quantities of interest. However, when
considering the time and frequency information concurrently, wavelets can be used to
determine the times and frequencies at which signal energy content is strongest, through
examination of scalograms and coscalograms, e.g. Gurley & Kareem (1999). More
specific insights into the linear and quadratic interplay between two signals in both time
and frequency can be gained utilizing wavelet coherence and bicoherence measures
(Gurley et al., 2003), as discussed in Chapter 6.
By exploiting the dual potential of wavelets, other analyses based primarily in
either the time or frequency domain can also be performed. For example, the evolution of
frequency content in time can be examined using wavelet scalograms and wavelet
instantaneous frequency spectra, as demonstrated in Chapter 5 for measured time
histories of earthquakes, wind velocity and wave elevation, known to possess nonlinear
3
and nonstationary features. Similarly, the distribution of wavelet coefficients with
frequency at a window in time provides a familiar spectral representation whose
evolutionary properties can be monitored to provide insights into nonlinear and
nonstationary behavior, as demonstrated with a number of measured and simulated time
histories in Chapter 7, which evaluates the performance of Wavelet Transforms and
Hilbert Spectral Analysis. Further, tracking the variation of wavelet coefficients in the
time domain permits complete system identification, a process that has particular
challenges for long-period, flexible structures, as discussed in Chapter 8.
Of course, the introduction of new analysis frameworks brings with it a number of
computational and processing challenges that must be fully reconciled to provide
physically meaningful results. A large component of this study is dedicated to exploring
and rectifying a number of these computational challenges, providing practical
processing tools and guidelines for the implementation of wavelet transforms,
particularly in the analysis of long-period systems. Although challenges did not surface in
prior applications concerned with mechanical systems, who are generally characterized
by higher frequency, broader-band signals, the transition to the time-frequency domain
for the analysis of Civil Engineering structures highlighted the need to understand more
fully various processing concerns, particularly for the popular Morlet wavelet.
Specifically, as these systems may possess longer period motions and thus require finer
frequency resolutions, the particular impacts of end effects become increasingly apparent.
The discussion of these issues begins in Chapter 3 with an overview of Wavelet
Transform theory and its relationships to the analytic signal. Chapter 4 then delves more
fully into the various processing concerns in the context of the wavelet’s multi-resolution
4
character and includes guidelines for selection of wavelet central frequencies, highlights
their role in complete modal separation, and quantifies their contributions to end effects
errors, which may be minimized through a reflective padding scheme. Other topics
addressed include the discretization of the time-frequency plane and ridge extraction
techniques. Further, these processing issues and analysis guidelines are articulated
throughout Chapters 5-8 in the analyses and examples that follow. For example, the
presence of statistical noise due to a lack of ensemble averaging in wavelet coherence
maps is discussed and rectified in Chapter 6 through a variety of thresholding techniques
and variable integration schemes. Though the application of wavelet transforms in Civil
Engineering is in its infancy, the examples provided in this study demonstrate its promise
as a tool to redefine the probabilistic and statistical analysis of processes in Civil
Engineering and beyond.
1.3 The Call to Full-Scale
As discussed in Chapter 2, constructed facilities provide the most viable venue for study
of inherent damping, motivating the need to develop extensive monitoring programs for a
diverse suite of buildings of varying height and structural form. The high-rise community
would particularly benefit from this enhanced understanding. The designers in this
community limit the drifts of a structure through an increase in stiffness provided by an
efficient choice of lateral system to satisfy serviceability criteria, all the while
recognizing that such modifications may not always significantly reduce the levels of
acceleration, of particular importance for occupant comfort concerns. In this latter case, it
is the structure’s damping that becomes the integral component in satisfying the crucial
5
habitability criteria for tall buildings, implying that a better understanding of the inherent
damping levels in common construction is necessary to insure that modern designs are
capable of providing the required provisions for energy dissipation. In fact, the high-rise
community’s need for a firm and reliable understanding of in-situ damping levels is
underscored by the plethora of auxiliary damping devices that have been installed in
recent decades (Kareem et al., 1999).
However, an enhanced understanding of inherent damping levels is not the sole
benefit of full-scale monitoring. Consider the fact that tall buildings are actually one of
the few constructed facilities whose design relies solely upon analytical and scaled wind
tunnel models that have yet to be systematically validated in full-scale. Therefore, if fullscale data could be compared against design predictions, tall building designers and wind
engineers would have a firmer understanding of the deficiencies and strengths of the
current design practice. Thus, to address these ranging needs, three tall buildings in
Chicago, representing structural systems most common to high-rise construction, are
being continuously monitored by the University of Notre Dame, in collaboration with the
Boundary Layer Wind Tunnel Laboratory at the University of Western Ontario and
Skidmore Owings and Merrill in Chicago (Abdelrazaq et al., 2000). The subsequent
analysis of the data collected through these efforts will provide valuable insight into a
variety of response characteristics for tall buildings, including the inherent damping, and
permit the systematic validation and enhancement of existing design practice through
comparisons with analytical and wind tunnel response estimates.
6
Traditional monitoring devices, including anemometers and accelerometers,
whose selection, assembly, and installation are discussed in Chapter 9, are supplemented
by high-precision Global Positioning Systems (GPS) to monitor structural displacements
that were previously difficult to recover, particularly the static components of wind–
induced response and thermal effects. The systems, with real time kinematic (RTK)
potential, allow for complete dynamic monitoring of displacements at up to 10
samples/second. Chapter 10 introduces the concept of Global Positioning Systems for
monitoring structural displacements in Civil Engineering and the anticipated sources of
error, while Chapter 11 details the configuration and analysis parameters for the GPS
components used in this research. Chapter 12 discusses a series of calibration tests
investigating the influence of satellite position on system performance and verifying the
ability of the GPS instrumentation to track complex waveforms and realistic long-period
structural response to random excitations for motions at the sub-centimeter level and for
frequencies up to 2 Hz. Chapter 13 then presents an analysis of some of the preliminary
data from this on-going full-scale monitoring program.
Although the tools developed in this study draw heavily from a multi-disciplinary
foundation, they are developed with the intent of improving the Civil Engineer’s ability
to completely characterize all the components of Equation 1.1, from the nonlinear and
nonstationary loads impacting the structure, to the nonlinear dynamic features in the
structure itself, none more illusive than inherent damping. As Chapter 14 summarizes,
this often requires excursions into new research areas, borrowing from other disciplines
in order to establish foundations for future advancements in Civil Engineering.
7
CHAPTER 2
DAMPING IDENTIFICATION AND ASSOCIATED UNCERTAINTY
2.1 Introduction
Despite the advancements that have been made in structural engineering in the last
century, one critical parameter remains an enigma: damping. The complexity of this
parameter is in part due to the diversity of sources contributing to the overall energy
dissipation capability. These sources include: 1) damping inherent to the bulk material of
which the system is formed, 2) boundary damping due to the dissipation between
interfaces or joints in the structure, and 3) dissipation associated with the structure’s
contact with soil or a fluid field, in the case of aerodynamic or hydrodynamic damping.
In particular, the boundary damping is difficult to quantify and represents significant nonlinear losses. It is unclear which variables affect the damping forces and the appropriate
models to represent these effects. As the choice of model relies on the mechanisms of
damping, the fact that there are numerous mechanisms, which are more varied and less
understood than the physical mechanisms governing mass and inertia, makes the
appropriate choice of model increasingly difficult (Woodhouse, 1999; Kijewski &
Kareem, 2000). As damping cannot be related to building materials and member
configurations in a direct manner like mass and stiffness, a universal model for structural
8
damping is not possible and damping levels are simply assigned in the design process
based on limited apocryphal data.
Consider this generalized form of the aforementioned equation of motion, for a
single-degree-of-freedom (SDOF) mechanical oscillator, given by
MX&& + f D + KX = F .
(2.1)
The linearity of this expression is achieved by replacing the damping force fD with a
viscous damping model:
f d ( x& ) = CX& ,
(2.2)
where C is the viscous damping coefficient. The result is the following equation of
motion
2
X&& + 2ξω n X& + ω n X = F
M
(2.3)
where ωn is the natural angular frequency (ωn = 2πfn) and ξ is the damping ratio, the ratio
of the assigned viscous damping to the critical damping value. Under this accepted
practice, any source of nonlinearity is obscured, as the linear viscous damping ratio is
inherently independent of amplitude, in contrast to other damping models, e.g. Coulomb
or quadratic (Kareem & Kijewski, 2000).
The shortcomings of assuming viscous damping levels became apparent with the
transition to light and flexible structures, as assumed levels of damping, on the order of
1% critical damping for steel structures and 2% for concrete, were often not realized in
the constructed building. For this class of tall, wind-sensitive structures, the diminished
9
levels of damping realized in practice led to a host of serviceability and especially
habitability problems that were not anticipated in design. The resounding difficulty in
engineering known levels of damping in design was partly responsible for the flurry of
auxiliary damping devices in recent years, which were found to provide measurable and
controllable levels of this critical parameter (Kareem et al., 1999).
Since an exact model for damping cannot be constructed on an element level, a
viable alternative was to develop empirical expressions for modal damping based upon
the levels observed in existing structures. Of course, this requires an extensive and
reliable survey of constructed facilities. Only recently have such databases become
available in the form of the Japanese Damping Data Base (JDDB) (Tamura et al., 1996)
featuring dynamic properties of 278 structures, and an international database featuring
dynamic properties of 185 buildings in Asia, Europe and North America (Lagomarsino &
Pagnini, 1995). From these and other databases, a variety of expressions have been
developed, including those by Engineering Sciences Data Unit (1990), based upon the
natural frequency and dimension of the building, and those fit to the JDDB dependent
upon the natural frequency of reinforced concrete structures (Tamura, 1997). Despite the
efforts to fit these existing databases with empirical functions, which correlate damping
with parameters such as natural frequency, it appears that the levels of damping to be
expected in the structure are a function of numerous other variables including the
material and foundation type. The situation is compounded by the fact that additional
levels of damping may be provided by non-structural elements and cladding components.
As these characteristics vary from structure to structure, the identification of an “expected
level of damping” for a given design becomes a daunting task.
10
The situation is further complicated by the fact that the full-scale damping data
currently cataloged manifests considerable scatter, indicating that significant errors may
be introduced. Haviland (1976) reported a wide range of data for different levels of
response amplitudes in a variety of structural systems and building heights and showed
that log-normal and gamma distributions provided the best fit to the damping variations.
The coefficient of variation (CoV) of the damping estimates, defined as the ratio of the
standard deviation σ to the mean µ, based on this data set, varied from 42% to 87%.
Davenport & Hill-Carroll (1986) reexamined the database and noted that the CoV ranged
from 33% to 78% and suggested a value of 40%. Thus the use of empirical expressions
based upon data with this degree of scatter could lead to the design of a building that on
paper may meet serviceability and occupant comfort criteria, but in practice, does not.
The limited success of these efforts may be due to a lack of structural diversity in the fullscale observations, the damping parameter’s observed amplitude dependence, and
difficulty in its estimation from measured ambient vibration data. This latter issue will be
addressed in greater detail throughout this chapter.
2.2 Traditional Approaches to Damping Identification from Wind-Induced
Response
As the analysis of measured response currently provides the only viable means to
quantify the levels of damping in structures, methodologies for structural testing,
monitoring and system identification have received considerable attention in recent
decades. Through these efforts, a host of methodologies for system identification have
surfaced, each tailored to specific applications. In the case where the system inputs are
11
measured or known, (e.g. forced vibrations using shaker, impact testing, earthquake
response), a variety of attractive system identification tools are available, e.g. Ghanem &
Shinozuka (1995). However, in the case of wind loading, due to spatio-temporal
variations in surface pressure, there is no direct relationship between the wind speed and
associated wind loads on the various faces of the structure (Simiu & Scanlan, 1996).
Thus, for the analysis of most full-scale data from wind-induced response, system
identification approaches requiring measured input are not viable. As a result, the
assumptions of Gaussian, stationary, white noise input are invoked to permit meaningful
analysis of wind-induced response by limited system identification approaches, e.g.
Desforges et al. (1995), Littler (1995), Wang & Haldar (1997). Among these approaches,
spectral analysis (SA) and the time-domain-based, Random Decrement Technique (RDT)
are the most common, with the spectral approach being the more traditional of the two.
2.2.1 Spectral Analysis and Inherent Errors
With the advent of the Fast Fourier Transform in the 1960’s, Fourier spectral analysis
became the primary means by which to analyze signals in Civil Engineering, as the
superposition of harmonic components in Fourier Analysis is intuitively appealing for
mechanical oscillators, providing a global energy distribution at each frequency f
contained in the signal x(t), i.e. producing a “stationary spectrum” Xˆ ( f ) called the
Fourier spectrum:
∞
Xˆ ( f ) =
∫ x(t )e
−∞
12
−i 2πft
dt .
(2.4)
Particularly in the case of wind-excited structures, the assumption of white noise input
yields an attractive representation of the unknown input as a constant power spectrum, an
assumption particularly valid in the narrow bandwidth of structural response. For this
reason, spectral analysis became the popular choice for system identification for of windinduced response.
As discussed, for example, in Bendat & Piersol (1986), the spectral estimate can
be simply generated by segmenting the measured response time history into blocks of
sufficient length T to provide the desired spectral resolution ∆fFFT, defined as the
difference between adjacent discrete frequencies, given by:
fk =
k
k
=
T N∆t
k= 0, 1, …, N-1
(2.5)
where N is the actual number of discrete data points xn in the block of length T, sampled
at the time step ∆t. Commonly, N is selected to the nearest power of 2 to permit use of the
Fast Fourier Transform (FFT), a convention used in this chapter, according to
N −1
⎡ 2πkn ⎤
Xˆ ( f k ) = ∆t ∑ xn exp ⎢− i
N ⎥⎦
⎣
n =0
k= 0, 1, …, N-1.
(2.6)
The FFT is applied to each block of data to produce the Fourier coefficients according to
Equation 2.6. The squared magnitudes of each set of Fourier coefficients produces a
single raw spectrum. The spectra produced by this method are then averaged, under
standard ergodic assumptions, to obtain an estimate of the power spectral density (PSD),
Sˆxx ( f ) evaluated at discrete frequencies, fk:
13
1⎛ 1
Sˆ xx ( f k ) = ⎜⎜
T ⎝ Ns
Ns
∑ Xˆ
nn =1
nn
2⎞
( f k ) ⎟⎟ k= 0, 1, …, N-1.
⎠
(2.7)
where Ns is the number of spectra being averaged.
The use of Fourier Transforms in this and other applications have highlighted
additional concerns such as side lobe leakage, often requiring additional windowing
measures that can increase the effective bandwidth of the spectra. As a result, an
understanding of spectral leakage and aliasing is requisite when applying the Fourier
Transform. These processing issues are discussed in more detail in (Bendat & Piersol,
1986).
The normalized bias error of the resulting power spectrum in the vicinity of the
resonance frequency fr reflects the importance of the frequency resolution achieved in
this process (Bendat & Piersol, 1986):
[
]
1 ⎛ ∆f
ε b Sˆ xx ( f r ) ≈ − ⎜⎜ FFT
3 ⎝ Br
⎞
⎟⎟
⎠
2
(2.8)
where Br is the half-power bandwidth of the Fourier power spectrum, approximated by
Br = 2ξfn.
(2.9)
Note the introduction of the half-power bandwidth (HPBW) in Equation 2.9 is an
approximation valid only for lightly damped systems, i.e. ξ < 0.1, for which the resonant
frequency may be taken as the natural frequency of the system (Bendat & Piersol, 1986).
Conveniently, this relationship permits a very simple and direct means of system
identification from the response PSD, given that the input spectrum is constant. Again,
14
this assumption is typically acceptable when the ambient excitations of the system can be
approximated as white noise, at least in the vicinity of the spectral peak. Figure 2.1
illustrates that the half-power bandwidth may be determined by identifying, via
interpolation between the discrete frequencies, the two frequencies f2 and f1 that
correspond to half the maximum amplitude of the PSD. Assuming symmetry of the
spectral peak, the HPBW is then defined as the difference between these two frequencies:
Br = f2- f1, with the frequency corresponding to the spectral peak taken as the natural
frequency of the system. This permits the system damping to be identified readily from
Equation 2.9. On the other hand, more sophisticated curve fitting the response spectra
may also be conducted in lieu of relying on the HPBW e.g. Littler (1995) or when limited
data is available, e.g. the maximum likelihood method (Montpellier, 1996) with
parameter averaging (Montpellier et al., 1998) and autoregressive (AR) modeling of
response spectra (Cao et al., 1997; Kijewski & Kareem, 1999).
These alternative schemes are useful in situations where limited amounts of data
render the generation of a smooth spectrum difficult, since it is governed by the limitation
of the normalized variance given in Bendat & Piersol (1986) as
[
]
1
var Sˆ xx ( f ) ≈
.
Ns
(2.10)
Note that this expression assumes that the raw spectra were determined from
independent, disjoint blocks of data of length T without the inclusion of windowing.
Overlapped processing of spectral estimates may be invoked when limited data are
available, as discussed in Bendat & Piersol (1986).
15
Sˆ xx ( f )
[
max Sˆ xx ( f )
[
]
1
max Sˆ xx ( f )
2
]
Br
f1 f2
fn
f
FIGURE 2.1. Schematic of half-power bandwidth
method of spectral system identification
X
For the narrowband processes common to Civil Engineering, small spectral
bandwidths place restrictions on the frequency resolution of the transformed data, as the
spectral resolution must be fine enough to resolve the sharp spectral peak, dictated by the
aforementioned bias error in Equation 2.8. In addition, to minimize the variance of the
spectral estimate in Equation 2.10, a sufficient number (Ns) of sample spectra must be
averaged. This introduces the infamous paradox in spectral analysis: considering that
only limited data are available for analysis, increasing the length of each ensemble to
reduce bias errors directly implies that there will be fewer ensembles available for
minimizing the variance. On the other hand, increasing the number of ensembles restricts
the length of each ensemble, increasing bias. The negative bias error implies that the
amplitude of the peak of the spectrum is always underestimated, leading to an
overestimation of damping. In the case of stationary data analysis, increasing the time
16
interval T over which the analysis is conducted will decrease the random errors in the
estimates of statistical properties; however in the case of nonstationary signals, an
increase in this interval T will result in a smoothing of the actual time variations in the
statistical properties of the signal, reiterating the importance of stationarity assumptions
in Fourier analysis. Thus the primary challenge in spectral analysis is obtaining a
sufficient amount of stationary data to minimize both competing sources of error. Note
that selective ensemble averaging and parameter averaging have been proposed to relax
the stringent stationarity requirements (Littler, 1995; Isyumov & Morrish, 1997).
2.2.2 Random Decrement Technique
One way to avoid entirely the resolution issues associated with Fourier Transform
techniques is to restrict analysis to the time domain. Though system identification in the
time domain is often difficult without explicit knowledge of the input, under the same
general assumptions made in spectral analysis for wind-induced response data, alternative
analysis tools are available. One popular approach is the Random Decrement Technique
(Gurley & Kareem, 1996), due to its ability to overcome the strict requirements for
lengthy stationary data imposed by traditional spectral approaches.
The decrement is generated by capturing a sample of prescribed length from the
time history upon the satisfaction of a threshold condition (Cole, 1973). This triggering
condition, in its strictest sense, will specify both amplitude and slope criteria, though
applications of the RDT in the literature have adopted varying trigger conditions, as
discussed in later in Section 2.2.2.4. Note also that the duration of segments captured is
17
subjective, but typically is on the order of a several cycles of oscillation. The segments
meeting these conditions are averaged to in essence remove the random component of the
response, assumed to be zero mean, leaving Random Decrement Signature (RDS).
Mathematically this operation is represented by an expectation
[
D(τ ) ≡ E X (t 2 ) X (t1 ) = X o I X& (t1 ) = X& o
]
(2.11)
where the assumption of stationarity allows the introduction of a time variable τ = t 2 − t1 .
The RDS was shown by Vandiver et al. (1982) to be proportional to the
autocorrelation signature ( Rxx (τ ) ) for the system, assuming in the derivation a more
relaxed trigger condition in that only an amplitude trigger Xo is specified:
D (τ ) ≡ E [X (t 2 ) X (t1 ) = X o ] = X o R xx (τ ) R xx (0) .
(2.12)
Though the RDT essentially provides an estimate of the autocorrelation function, it is
able to produce this estimate without the same strict requirements for lengthy stationary
data, as discussed by Jeary (1992).
The expectation in Equation 2.11, assuming ergodicity, can be replaced by an
average of Nr triggered segments Xtt(t) over a single time history
D(τ ) =
1
Nr
∑ (X
Nr
tt =1
tt
(τ ) X tt (0) = X o I X& tt (0) = X& o ).
(2.13)
A conceptualization of the process of generating the RDS, shown in Figure 2.2, reflects a
common representation of the RDT, in which the total response is viewed as the
superposition of the forced vibration response with the homogeneous component or free
18
Xtotal
X(t) | Xo, X& o
=
=
+
+
+
+
+
+
=
+
Σ
+
XF
+
Σ
+
+
Σ
+
=
+
FIGURE 2.2. Conceptualization of the Random Decrement
Technique for local extrema triggering condition
vibration decay from given initial conditions, i.e. the triggering condition. Successive
averaging of segments of this response will tend to cancel out the random component of
the response, assumed to be zero mean. This leads to the perception that the Random
Decrement Technique yields the free vibration response of the system. However, this is
only the case for a linear oscillator excited by Gaussian, zero-mean, white noise, for
which the autocorrelation function, normalized by a constant C, takes the following
form:
Rxx (τ ) = Ce −ξωnτ cos(ω Dτ ) ,
19
(2.14)
which is analogous to the free vibration response of an oscillator with damped natural
frequency ωD, assumed equivalent to its undamped counterpart for a lightly damped
system. As long as the white noise assumption remains valid (implications of this are
discussed in Kareem & Gurley (1996) and Spanos & Zeldin (1998)), the analogs between
Equations 2.11 and 2.14 may be exploited for system identification by a number of
strategies, e.g. via least squares minimization to obtain best-fit estimates of damping ξ
and natural frequency fn, letting C = Xo/Rxx(0).
Alternatively, the damping can easily be identified from the peaks in the decay
curve via the logarithmic decrement. Let Dp(n) and Dp(n+1) define two consecutive
peaks in the RDS decay curve. The logarithmic decrement (LogDec) is then defined from
these two peaks:
δ ≡ ln
D p ( n)
D p (n + 1)
=
2πξ
1−ξ 2
,
(2.15)
implying that for lightly damped systems, δ ≈ 2πξ . It has been observed (Clough &
Penzien, 1993), that a more accurate estimate of the damping using logarithmic
decrement is obtained by considering peaks separated by m (multiple) cycles, yielding the
following expression for estimating the damping of lightly damped systems by
logarithmic decrement
ξ≈
D p ( n) − D p ( n + m)
2mπD p (n + m)
.
to directly identify the damping from the peaks in the decay curve.
20
(2.16)
Under the aforementioned theoretical assumptions, the RDS will be unbiased with
variance that can be expressed by (Vandiver et al., 1982):
var[D(τ )] = E[ D 2 (τ )] − E[ D(τ )]2 = Rxx (0) / N r [1 − Rxx2 (τ ) / R xx2 (0)] .
(2.17)
The presence of noise was ignored in this idealized derivation, as was the potential
correlation between the captured segments. Thus, to satisfy these theoretical constraints,
captured segments are often not allowed to overlap. Note also that, as indicated by
Equation 2.17, the amplitude level selected for the triggering condition is theorized to
have no direct effect on the variance, though it will dictate the number of segments
averaged. Clearly, this number of segments should be increased as much as possible to
minimize variance; however, in application, the amplitude of the trigger has been
observed to influence the accuracy of the resulting damping estimates, as discussed in
Section 2.2.2.2.4, and in the case of systems with amplitude-dependent damping, the
damping estimated from the resulting random decrement signature corresponds to the
viscous damping ration at the amplitude level associated with that trigger (Tamura &
Suganuma, 1996).
The preceding discussion assumed that the response was comprised of a single
mode. Conventionally, the incorporation of bandpass filtering (Kijewski & Kareem,
1999; Tamura & Suganuma, 1996) prior to the application of the RDT is required to
insure this condition for measured data. However, the Random Decrement Technique can
also be applied directly to a time history that may have multiple modes present. In this
case, the resulting RDS will be proportional to a multi-degree-of-freedom (MDOF)
autocorrelation function, or under the assumed conditions discussed previously, a
21
superposition free vibration decays for several modes. The identification of the MDOF
signature can then be conducted by an appropriate technique, e.g. Ibrahim Time Domain
Method (Ibrahim, 1977; Mahmoud et al., 2001) or, as discussed in Chapter 8, wavelet
transforms can be used to decouple the signature’s individual components.
2.2.2.1
Types of Triggering Conditions
The use of RDT in the literature has highlighted a differing opinion on the number of
segments required to successfully average out the random component of the response. As
demonstrated later in this chapter, this may significantly depend on the level of
randomness in the excitation itself. However, the differing opinions may be more directly
attributed to the triggering condition employed. As shown in Equation 2.11, trigger
conditions can strictly specify either an amplitude or slope level or both, leading to a
variety of trigger conditions in the literature. Ibrahim (2001) discusses the four most
common triggers:
{X tt (0) = X o }
level crossing:
{X < X (0) < X }
{X (0) = 0 I X& (0) > 0}
positive point:
low
zero up-crossing:
local extrema:
n
tt
{X
(2.18)
low
(2.19)
high
(2.20)
tt
}
< X tt (0) < X high I X& tt (0) = 0 .
(2.21)
The level crossing condition is arguably the most fundamental triggering condition and is
assumed in the theoretical derivations in Vandiver et al. (1982) resulting in Equations
2.12 and 2.17. The positive point condition, which specifies a range of values in order to
generate more eligible samples, is a potentially more realistic version of the level
22
crossing criteria, as it is rare to encounter values within the time history that identically
equal some arbitrary trigger amplitude. The positive point condition, in order to more
strictly approach the level crossing condition, may assume its range based on some
percentage of the desired amplitude level, e.g. on the order of a few percent. On the other
hand, more robust ranges can be selected in Equation 2.19 to include a wide variety of
amplitude levels, akin to the procedure in Tamura & Suganuma (1996). The ranges were
selected in that study to guarantee that each had a pre-determined number of eligible
segments. The zero up crossing condition identifies zero crossing points within the signal,
though this condition again may require some relaxation to include a range of values
within a few percent, as it again is rare to encounter identically zero values in a sampled
time history. Note that this trigger condition has no potential to reference the signature to
a specific amplitude level and therefore lacks physical significance for the applications in
this study. The final condition, which is formally presented in Tamura & Suganuma
(1996), captures only peaks falling within a specified amplitude range. This triggering
condition is the most strict, as it firmly specifies both an amplitude and slope. This range
can be made very narrow to stringently reference a particular amplitude for tracking
amplitude-dependent dynamic properties, as advocated herein, or again relaxed for a
more robust range of amplitudes. Tamura & Suganuma (1996) utilized this relaxed
trigger to track amplitude-dependent damping in full-scale data. Note the trigger
condition more completely supported by the mathematical theory developed in Vandiver
et al. (1982) is the level crossing condition or positive point trigger with narrow range.
The local extrema approach merely becomes a special case of the level crossing
23
condition, reducing the number of eligible segments by enforcing the zero slope
condition.
To this suite of triggering conditions, one may also supplement the level crossing
by a positive slope. The best rationale for this trigger comes from the fact that, as the
signal ascends toward a local extremum, it may pass through the specified amplitude
level, and it may be assumed that, following this peak in the time history, the signal will
descend toward the zero level, again crossing the specified amplitude but with a negative
slope. As these events are usually within a limited time frame, segments captured at both
of these crossings will likely have a high degree of correlation. As the segments are
assumed independent in theory, the selection of a positive slope becomes a simple way to
prevent the capture of highly correlated segments. However, if other measures are being
taken to enforce the condition of nonoverlapping segments, this added slope condition is
truly not necessary.
2.2.2.2
Investigations of Factors Influencing RDT Quality
The sensitivity of RDS to the violation of various assumptions in its theory will be
explored in the subsequent sections. Authors have discussed the implications of violating
the white noise assumption in the RDT (Kareem & Gurley, 1996; Spanos & Zeldin,
1998), illustrating that the RDS cannot be equal to the free vibration curve if the
excitation is not truly white. However, there has been no treatment of the implications of
correlation between samples. Similarly, the influence of trigger condition and
nonstationarity, also not explored systematically in the literature, will be investigated in
24
the subsequent sections. For demonstrative purposes, analyses of a SDOF oscillator
excited by Gaussian, zero-mean, white noise are utilized. The response of this 0.2 Hz
system with 1% critical damping ratio was simulated for 12 hours, sampled at 2 Hz.
Unless stated otherwise, the strictest trigger condition of local extrema was utilized,
specifying an amplitude range of +/- 3% of the desired trigger amplitude.
One of critical parameters to be explored is the variance of the RDS. Though the
variance of the signature may be obtained directly from the Nr segments used in the
averaging process, this is merely a sample variance. However, resampling schemes such
as bootstrapping can be exercised as an alternative means by which to estimate the true
variance of the signatures (Efron & Tibshirani, 1993). A bootstrap resampling scheme
will be utilized to gauge the variance of the random decrement signature, so as to obtain a
better estimate of the true variance. The Appendix of this study contains more
information on the bootstrap methodology for estimating the variance of random
decrement signatures and the simulation of variance envelopes, which highlight the areas
of greatest variance in the random decrement signature. It is hoped that the introduction
of such a scheme will provide practitioners with a simple means by which to estimate the
variance of their RDS and provide a measure of the reliability when theoretical
assumptions are not entirely met and closed form expressions for variance cannot be used
due to violations of the assumptions made in their derivation. All bootstrapped estimates
discussed in the subsequent sections are based on B=50 replicates.
25
2.2.2.2.1 Suggested Expression for Random Decrement Signature Variance
Before analyzing the various factors contributing to the performance of the Random
Decrement Technique, the bootstrap variance estimate will validated against the closed
form expression for variance in Equation 2.17. The examination of the simulated data
shown in Figure 2.3 agrees in trend with the closed form expression. As the initial
conditions are indeed enforced at τ=0, the RDS is theoretically most reliable at this point.
Subsequently, the signature increasingly breaks down further from this point, as the
deviations between the theoretical decay curve and the RDS demonstrate. The
bootstrapped replicates of the decrement signature form a variance envelope, shown as
the second of a sequence of plots in Figure 2.3. This provides some appreciation of the
variance in estimates, propagating with time.
Note also that the closed form expression for variance in Equation 2.17 is based
upon the autocorrelation function in Equation 2.14 and yields results that are counterintuitive to the actual behavior. In theory, the variance will oscillate as a squared cosine
function, indicating that it will reach maximum every half cycle. With time, these
oscillations diminish and the variance will approach a near constant value of the signal
variance divided by Nr. However, this variance estimate would indicate that within a half
cycle of the trigger threshold the error has already reached its maximum value. This
would indicate that in its present form, the error estimate is not a true representation of
the behavior of the decrement, as the bootstrapped estimates and intuitive arguments
presented in Vandiver et al. (1982) also confirm. This work offers a more practical,
limiting case for the error
26
RDS
Var.
Envelope
Var.,
Bootstrap
Var.,
Theory
Time [s]
FIGURE 2.3. (top to bottom) RDT against theoretical autocorrelation,
bootstrap variance envelope, bootstrapped variance estimate against
Equation 2.17, theoretical variance in Equation 2.13, uncorrelated
segments
ε RDS (τ ) = R XX (0) / N r (1 − e −2ξω
Dτ
).
(2.22)
As shown by the dashed line in Figure 2.3, this limiting error forms the lower
bound in theory, providing a more reasonable approximation of RDS variance. The
bootstrap, in providing a realistic estimate of variance also fits well against Equation
2.22. As the bootstrap variance is consistent with the theoretical variance, it is deemed
appropriate for variance estimation in this chapter.
27
2.2.2.2.2 Significance of Overlap Between Captured Segments
In the development of the random decrement variance in Equation 2.17, it was assumed
that the segments captured for averaging are independent, i.e. they do not share any part
of the time history. This restriction limits the amount of segments available for averaging,
an especially concerning issue since the number of segments averaged is a critical
parameter in reducing the variance of random decrement signatures, as will be shown in
the subsequent section. Therefore, the ability to allow overlap in captured segments
would be beneficial. To explore the implications of violating this assumption,
bootstrapping is used to estimate the variance of the random decrement signatures when
overlapping of segments is permitted.
By comparing Figure 2.3 and 2.4, the effects of correlation on the quality of the
estimates appears not to be considerable, in comparison to the same analysis for
uncorrelated segments in Figure 2.3. In fact, aside from some random fluctuations in the
bootstrapped variance estimate, the correlated case produces the same limiting variance
as the uncorrelated case, fit by Equation 2.22. Further, by examining Figure 2.5, one can
see that there is no increase in the bootstrapped estimate of cyclic variance (averaged
bootstrap variance in the RDS over one cycle of oscillation) as a result of permitting
some correlation between samples. By allowing some overlap between adjacent samples,
the user is now afforded additional samples for averaging, a critical requirement for the
use of RDT.
28
RDS
Var.,
Bootstrap
Var.,
Theory
Var.
Envelope
Time [s]
FIGURE 2.4. (top to bottom) RDT against theoretical autocorrelation,
bootstrap variance envelope, bootstrapped variance estimate against
Equation 2.12, theoretical variance in Equation 2.13, correlated
segments
2.2.2.2.3 Required Number of Captured Segments
Despite the widespread use of the RDT, there still is much debate surrounding the
amount of data required to yield reliable estimates. In order to accomplish the averaging
required to completely remove the random component of the response, some researchers
recommend 400 to 500 segments (Yang et al., 1983), while others recommend at least
2000 (Tamura et al., 1992).
As shown by Figure 2.5, by averaging the bootstrap
variances over each cycle of the RDT signature, one can monitor the increase in variance
as more cycles are considered in the estimate. Clearly, as the number of segments
29
averaged increases, this variance decreases. However, it is interesting to note whether or
not the RDS variance is a good indication of the accuracy of a given damping estimate.
The error of logarithmic decrement estimates averaged over a number of cycles is also
shown in Figure 2.5. The reliability of single cycle damping estimates is poor even
though this is where the variance in the RDT estimate is minimum. Thus, a more accurate
means of damping estimation is required to permit estimates using only the first few
cycles of the RDS, which are more stable. In fact, only by considering 3 or more cycles
do estimates approach an accuracy of 10%; however, there is a trade-off in that the
variance in the estimates also increases with the number of cycles. Thus the estimates of
damping by logarithmic decrement are best when performed within the first 4 cycles with
more than Nr=200 averages, when the local extrema triggering condition is utilized, for
which errors in damping estimation by logarithmic decrement are between 10-20%. Later
examples will further demonstrate the accuracy of damping estimation including the
influence of trigger condition on the required number of averages. Finally, to enhance the
quality of damping estimate over the early cycles of the RDS, the use of a least squares
curve-fitting approach may be advocated, as discussed later in this chapter.
2.2.2.2.4 Significance of Triggering Condition on Damping Estimation
The last section reflected that, for a given amplitude, an increase in the number of
segments averaged in the decrement will reduce the signature variance, as predicted by
the expression in Equation 2.17, though the performance in terms of damping estimation
can be quite variable. Certainly, the variance of the RDS signifies the potential variability
in the signature. As the damping is identified solely from the peaks of the function
30
Avg. Cyclic Variance [%]
Error in Damping Estimate [%]
Number of Cycles
Number of Cycles
FIGURE 2.5. RDT cyclic variance (top) and resulting damping
estimation error
31
marking the decay curve of the RDS, although variance of the signature on the whole
may be small, damping values can still have some scatter if these peaks are the least
reliable components of the RDS, justifying the findings in Figure 2.5. Further, the peaks
in the decay curve may be less than or greater than the amplitudes expected in theory,
leading to damping values that can be over or underestimates. For spectral analysis, the
spectral bandwidth is always wider than theory, leading to a consistently overestimated
and biased damping estimate. Thus the behavior of estimated damping values, as shown
in Figure 2.5, must be investigated, along with the influence of trigger condition, trigger
amplitude and overlap between the decrement segments. Such an investigation follows
herein.
The only unique trigger condition that allows some amplitude referencing is the
level crossing or positive point condition, with the local extrema being a special case of
this condition. These latter two triggers are explored for a host of amplitude levels, with a
range of +/-3% of the specified amplitude level. Recall that the theoretical developments
of the RDT in Equations 2.12 and 2.17 assume the captured segments don’t overlap. Thus
three correlation cases are considered by varying the length (TRDS) of the captured
segments to investigate the influence of this assumption. These correlation conditions are
TRDS = 50 seconds (10 cycles of oscillation), implying that segments 50 seconds long will
be captured and no portion of the time history within those 50 seconds is eligible to
trigger the capture of a new segment. A second case lengthens this requirement to
TRDS = 100 seconds (20 cycles of oscillation), reducing further the potential for
correlation between segments. A third case places no restriction on overlapping
segments, thus TRDS = 0 s, allowing correlation freely.
32
The random response of the system discussed in Section 2.2.2.2 was simulated
four times (referred to as Runs I, II, III, IV) using different seeds in the random number
generator to demonstrate the variability of response histories (shown in Figure 2.6) with
the same spectral and probabilistic description, shown in Figure 2.7, distributing
Gaussian about a zero mean with standard deviations listed in Table 2.1. The power
spectral densities of the response all identify the frequency of the oscillator and all appear
to have the same distribution of energy in the low frequencies. While Runs II and IV
have higher energy levels in the high frequencies, this characteristic has no obvious
influence on results.
x(t)
Run II
x(t)
Run I
Run III
x(t)
x(t)
Run IV
t [s]
t [s]
FIGURE 2.6. Time histories of SDOF response for Runs I-IV
TABLE 2.1
STANDARD DEVIATIONS OF SIMULATED TIME HISTORIES
Run I
3.11
Run II
3.00
Run III
2.99
33
Run IV
3.17
No. of observations
Run I
Run II
Run II
Run III
Run IV
Xo
Xo
Run
IV
Run III
Xo
Run I
Xo
FIGURE 2.7. Histograms (left) and power spectra for simulated response in Runs
I-IV
The Random Decrement Technique is applied to each simulated time history,
using varying trigger conditions and the three correlation conditions described above. A
damping estimate is then obtained from two full cycles of the resulting signatures using
the logarithmic decrement. To verify the inherent error in the LogDec approach, damping
was estimated from the first two cycles of impulse response functions, generated directly
from the system equation of motion. It was found that the LogDec overestimated the
damping of this system as 0.0107. As a result, the quality of LogDec damping estimates
from the RDS should be appropriately weighted for this inherent bias. The tabulated
results provided in this chapter therefore shade the values of damping which fall within
10% of the LogDec bias, i.e. 0.0096-0.0117, in darker hues. Results corresponding to 1120% error relative to the biased LogDec estimate, i.e. damping values of 0.0085-0.0095
and 0.0118-0.0128, are shaded with a lighter hue.
34
Table 2.2 lists the results for the positive point trigger, specifying the target
amplitude as a multiple M of the standard deviation of the response. Nrtot is the total
number of segments satisfying the trigger condition, while Nr is the number of segments
actually averaged to form the RDS, after the enforcement of correlation restrictions. A
visual comparison seems to reflect that Run II shows the poorest performance,
demonstrating that the randomness inherent in some records may require relatively a
larger number of segments to fully average out, as will be discussed further in Section
2.3. For the positive point trigger, allowing correlation understandably increases the
number of samples in the average, yielding more reliable damping estimates. Therefore,
the influence of correlation does not appear negative. A fact also affirmed in Section
2.2.2.2.2. The damping appears to be consistently estimated in Runs I, III and IV, when a
the trigger satisfies the following condition: 3σ > Xo > 1σ.
Low triggers, though having a large number of samples in the average, yield
inconsistent performance. This is an interesting result, as the variance of the decrement
signature is shown to minimize with the number of averages and lower trigger amplitudes
certainly afford more potential averages in the Gaussian distribution of amplitude values.
Although variance of decrement signatures is theorized not to depend upon the trigger
level, the performance is inconsistent for Xo < σ, though at times the estimates are quite
accurate. Intuitively, one may argue that the trigger condition bears some influence on the
performance of the decrement. As noted by Vandiver et al. (1982), triggers below or on
the order of the root mean square (RMS) of the process will occur frequently within the
time history, increasing the potential for correlation that may lead to larger variance in
cases where TRDS = 0 s. However, simulations show that the greater number of averages
35
TABLE 2.2
ESTIMATED LOGDEC DAMPING VALUES USING POSITIVE POINT
TRIGGERS
ΤRDS = 0 s
ξ
0.0099
0.0146
0.0094
0.0100
0.0096
0.0101
0.0106
0.0079
0.0074
M
0.5
0.75
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Xo
1.55
2.33
3.10
4.66
6.21
7.76
9.32
10.8
12.4
Nr/Nrtot
902/903
1198/1202
1251/1251
966/968
552/555
250/252
64/64
16/16
3/3
M
0.5
0.75
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Xo
1.50
2.25
3.00
4.50
6.00
7.50
9.00
10.5
12.0
ΤRDS = 0 s
Nr/Nrtot
ξ
882/885
0.0094
1223/1224
0.0182
1216/1216
0.0148
1008/1008
0.0118
539/539
0.0103
248/248
0.0126
77/77
0.0126
12/12
0.0108
2/2
0.0042
M
0.5
0.75
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Xo
1.49
2.24
2.99
4.49
5.98
7.48
8.98
10.5
11.9
ΤRDS = 0 s
Nr/Nrtot
ξ
897/899
0.0106
1141/1142
0.0105
1232/1232
0.0077
992/992
0.0104
542/542
0.0115
183/183
0.0101
76/76
0.0108
21/21
0.0050
6/6
0.0128
M
0.5
0.75
1.0
1.5
2.0
2.5
3.0
3.5
4.0
X0
1.59
2.38
3.17
4.76
6.34
7.93
9.52
11.1
12.7
ΤRDS = 0 s
Nr/Nrtot
ξ
922/923
0.0131
1108/1110
0.0110
1239/1239
0.0088
1043/1043
0.0104
553/553
0.0100
220/220
0.0094
95/95
0.0101
24/24
0.0141
0/0
N/A
Run I
ΤRDS = 50 s
Nr/Nrtot
ξ
432/903
0.0123
489/1202
0.0162
497/1251
0.0133
966/968
0.0100
212/555
0.0089
99/252
0.0103
30/64
0.0107
5/16
0.0029
1/3
0.0028
Run II
ΤRDS = 50 s
Nr/Nrtot
ξ
426/885
0.0086
510/1224
0.0168
489/1216
0.0113
385/1008
0.0124
225/539
0.0100
98/248
0.0141
31/77
0.0119
7/12
0.0145
1/2
0.0055
Run III
ΤRDS = 50 s
Nr/Nrtot
ξ
428/899
0.0083
480/1142
0.0116
488/1232
0.0046
385/992
0.0128
222/542
0.0116
80/183
0.0120
32/76
0.0087
8/21
0.0057
4/6
0.0136
Run IV
ΤRDS = 50 s
Nr/Nrtot
ξ
432/923
0.0137
469/1110
0.0093
473/1239
0.0106
386/1043
0.0112
211/553
0.0106
93/220
0.0089
38/95
0.0111
11/24
0.0162
0/0
N/A
36
ΤRDS = 100 s
Nr/Nrtot
ξ
291/903
0.0145
313/1202
0.0191
317/1251
0.0091
256/968
0.0123
157/555
0.0086
81/252
0.0107
27/64
0.0120
5/16
0.0029
1/3
0.0028
ΤRDS = 100 s
Nr/Nrtot
ξ
280/885
0.0072
322/1224
0.0195
312/1216
0.0113
268/1008
0.0135
168/539
0.0146
78/248
0.0144
28/77
0.0111
7/12
0.0145
1/2
0.0055
ΤRDS = 100 s
Nr/Nrtot
ξ
287/899
0.0126
313/1142
0.0144
312/1232
0.0045
253/992
0.0097
168/542
0.0113
66/183
0.0125
27/76
0.0088
8/21
0.0057
4/6
0.0136
ΤRDS = 100 s
Nr/Nrtot
ξ
283/923
0.0175
300/1110
0.0097
304/1239
0.0175
272/1043
0.0103
160/553
0.0099
76/220
0.0096
33/95
0.0096
11/24
0.0162
0/0
N/A
afforded offsets any increase in variance due to correlation, though it may be theorized
that the lower amplitude segments within the time history have a greater level of
correlation than the segments with higher amplitude, though this is yet to be proven.
Conversely, the large amplitude triggers may be assumed to be particularly unreliable due
to the limited number of averages.
The range of behavior in Table 2.2 further emphasizes that specific features
within a given time history may impact performance, even when all the very idealized
theoretical conditions are satisfied, as typified by the poor performance of Run II.
Realizing that full-scale data is far less ideal, Jeary (1992) proposed some guidelines for
removing portions of time histories that may cause break down in decrement signatures,
“eliminating the type of data which produce spurious results.” As certain characteristics
in the time series can make the averaging process more difficult, Jeary recommends that
following types of sequences be omitted from the time histories:
•
“sequences in which “drop-outs are present”
•
“sequences in which sudden changes in [root mean square] rms occur”
•
“sequences in which beating between two modes of vibration occurs”
The customary practice of filtering about one mode of vibration aids particularly in
enforcing this last condition. However, Jeary advocates repeatability of results as well,
since spurious findings tend to distribute themselves randomly, in accordance with the
nature of the process. In this work, it is suggested that such repeatability may be achieved
by selecting several trigger amplitudes close to one another, identifying the damping
37
from each resulting decrement signature, and then examining the variability in damping
estimates. The variability is assumedly small due to the proximity in amplitude level and
the fact that the number of averaged segments should be similar. Outlier samples may
then be distinguished or more appropriately, an average of the identified properties over
this range can be taken as the damping estimate referenced to the average trigger, as
shown in Table 2.3.
TABLE 2.3
REPEATABILITY AND LOCAL AVERAGING FOR RUN I, LOCAL
EXTREMA TRIGGER, ΤRDS = 0
Xo
9.0
9.1
9.2
9.3
9.4
9.5
Nr
68
60
52
46
47
39
ξ
0.0088
0.0101
0.0117
0.0110
0.0096
0.0103
µ[Xo]
µ[ξ]
9.25
0.0102
Table 2.4 shows the results for the application of the RDT using the local extrema
trigger, where the same amplitude levels used in the positive point triggers in Table 2.2
are used as the amplitude levels in the local extrema trigger condition in Table 2.4. In
addition, as the mean peak value is often chosen as a potential trigger, since it will afford
a large number of averages, the analysis is conducted for this trigger as well. Looking at
the results for the four cases in Table 2.4, there is not much loss of samples due to the
enforcement of any correlation conditions, in part because the additional criteria on the
slope reduces the potential proximity of captured segments that would introduce
38
TABLE 2.4
ESTIMATED DAMPING VALUES USING LOCAL EXTREMA TRIGGER
Xo
1.55
2.33
3.1
4.66
6.21
7.76
9.32
10.8
12.4
3.74*
ΤRDS = 0 s
Nr/Nrtot
ξ
137/137
0.0226
223/223
0.0197
345/345
0.0131
344/344
0.0118
265/266
0.0124
147/148
0.0093
48/47
0.0098
14/14
0.0086
1/1
0.0028
379/379
0.0112
Xo
1.50
2.25
3.00
4.50
6.00
7.50
9.00
10.5
12.0
3.51*
ΤRDS = 0 s
Nr/Nrtot
ξ
130/130
0.0358
270/270
0.0314
345/345
0.0217
409/409
0.0145
272/272
0.0107
153/153
0.0129
52/52
0.0144
9/9
0.0122
2/2
0.0042
363/364
0.0162
Run I
ΤRDS = 50 s
Nr/Nrtot
ξ
137/137
0.0226
223/223
0.0197
345/345
0.0131
343/344
0.0117
265/266
0.0124
147/148
0.0093
45/47
0.0097
9/14
0.0081
1/1
0.0028
377/379
0.0113
Run II
ΤRDS = 50 s
Nr/Nrtot
ξ
130/130
0.0358
270/270
0.0314
345/345
0.0217
409/409
0.0145
272/272
0.0107
153/153
0.0129
49/52
0.0144
7/9
0.0145
1/2
0.0055
362/364
0.0164
ΤRDS = 100 s
Nr/Nrtot
ξ
137/137
0.0226
223/223
0.0197
345/345
0.0131
343/344
0.0117
265/266
0.0124
144/148
0.0098
44/47
0.0100
9/14
0.0081
1/1
0.0028
376/379
0.0115
ΤRDS = 100 s
Nr/Nrtot
ξ
130/130
0.0358
270/270
0.0314
345/345
0.0217
409/409
0.0145
272/272
0.0107
153/153
0.0129
48/52
0.0145
7/9
0.0145
1/2
0.0055
359/364
0.0164
*mean of peak values.
correlation. However, as the peak condition removes many segments that were eligible in
the positive point condition, the number of segments is minimized. Interestingly, when
capturing only segments initiating with peaks, adding more samples does not improve
performance dramatically and may even deteriorate the RDS. This is due to the fact that
over the three correlation conditions the number of samples being averaged does not vary
significantly enough to make dramatic impact on performance. As shown in Figure 2.4,
39
TABLE 2.4 (CON’T)
Xo
1.49
2.24
2.99
4.49
5.98
7.48
8.98
10.5
11.9
3.53*
ΤRDS = 0 s
Nr/Nrtot
ξ
131/131
0.0420
235/235
0.0172
309/309
0.0069
358/358
0.0092
281/281
0.0140
105/105
0.0125
46/46
0.0114
12/12
0.0025
3/3
0.0123
348/348
0.0129
Xo
1.59
2.38
3.17
4.76
6.34
7.93
9.52
11.1
12.7
3.44*
ΤRDS = 0 s
Nr/Nrtot
ξ
132/132
0.0541
230/231
0.0242
323/323
0.0139
395/395
0.0122
273/273
0.0128
126/126
0.0100
63/63
0.0100
20/20
0.0136
0/0
N/A
327/327
0.0131
Run III
ΤRDS = 50 s
Nr/Nrtot
ξ
47/131
0.0200
235/235
0.0172
309/309
0.0069
358/358
0.0092
279/281
0.0138
102/105
0.0118
44/46
0.0111
11/12
0.0034
2/3
0.0085
348/348
0.0129
Run IV
ΤRDS = 50 s
Nr/Nrtot
ξ
132/132
0.0541
230/231
0.0242
323/323
0.0139
394/395
0.0122
273/273
0.0128
124/126
0.0100
58/63
0.0096
12/20
0.0149
0/0
N/A
327/327
0.0131
ΤRDS = 100 s
Nr/Nrtot
ξ
129/131
0.0428
235/235
0.0172
309/309
0.0069
358/358
0.0092
278/281
0.0138
102/105
0.0118
43/46
0.0116
11/12
0.0034
2/3
0.0085
347/348
0.0131
ΤRDS = 100 s
Nr/Nrtot
ξ
132/132
0.0541
230/231
0.0242
323/323
0.0139
394/395
0.0122
272/273
0.0130
123/126
0.0100
57/63
0.0098
12/20
0.0149
0/0
N/A
327/327
0.0131
*mean of peak values.
despite general trends, the variance estimates can still have fluctuations, and this is
reflected in the results in Table 2.4 in the absence of significant increases in Nr. The
performance of the local extrema RDS tends to stabilize as more samples are added, as
shown later in Section 2.4.2. Thus if only limited data is available, the positive point
method may be better. Interestingly, the amplitude levels that again performed the best
were the same trigger levels that performed well in the positive point method. This
finding is again surprising. While a trigger in the vicinity of the peak distribution’s
40
median yields a large number of averages, higher amplitude trigger levels associated with
the 1.5-3σ amplitude often outperform them. This may again indicate that contrary to
Expression 2.17, the trigger amplitude has some influence on performance, in
conjunction with number of averages. It is arguable that the segments associated with
triggers in the amplitude range of 1-3σ may have correlation behaviors different from its
counterparts producing equal number of eligible segments at lower amplitudes.
In a final series of simulations, the trigger condition is relaxed to allow
consideration absolute values of a specified amplitude in Equations 2.18, 2.19 or 2.2.1.
This modified implementation of the decrement is given by
D(τ ) =
1
Nr
∑ (sgn( X
Nr
tt =1
tt
(0)) X tt (τ ) X tt (0) = X o I X& tt (0) = X& o
)
(2.23)
which should afford additional segments for averaging, though potentially raising
correlation concerns. The results from this investigation are shown in Table 2.5 and only
Runs I and II are considered for brevity. By allowing the potential for full correlation
(TRDS = 0) and using the positive point trigger, there is a marked increase in the number of
averages in comparison to Table 2.2, but with no significant improvement in
performance. Thus, at some point, segments may overlap so much that they do not help to
further reduce variance, though apparently causing little determent. In this case, enforcing
TRDS = 50 s seems to improve results for the low amplitude triggers, perhaps minimizing
the effects of correlation, but detracts from the performance of higher amplitude triggers
by reducing Nr. However, in the local extrema trigger simulations, the use of an absolute
value amplitude did appear to improve performance in the fully correlated and TRDS =
41
TABLE 2.5
INFLUENCE OF ABSOLUTE VALUE TRIGGER CONDITIONS
M
0.5
0.75
1.0
1.5
2.0
2.5
3.0
3.5
4.0
M
0.5
0.75
1.0
1.5
2.0
2.5
3.0
3.5
4.0
X0
1.55
2.33
3.10
4.66
6.21
7.76
9.32
10.8
12.4
X0
1.50
2.25
3.00
4.50
6.00
7.50
9.00
10.5
12.0
Abs. Value Amplitude Trigger
Run I
ΤRDS = 50 s
ΤRDS = 0 s
tot
Nr/Nrtot
N
/N
ξ
ξ
r
r
1835/1838 0.0090
568/1838
0.0107
2355/2361 0.0140
612/2361
0.0114
2492/2492 0.0095
602/2494
0.0115
1902/1907 0.0096
432/1907
0.0107
1143/1149 0.0098
253/1149
0.0087
502/505
0.0101
119/505
0.0078
128/128
0.0110
40/128
0.0098
38/38
0.0057
8/38
0.0077
5/5
0.0089
1/5
0.0028
Run II
ΤRDS = 50 s
ΤRDS = 0 s
tot
Nr/Nr
Nr/Nrtot
ξ
ξ
1816/1820 0.0101
562/1820
0.0070
2402/2404 0.0155
621/2404
0.0112
2497/2500 0.0141
614/2500
0.0135
2003/2003 0.0117
463/2003
0.0124
1109/1109 0.0113
278/1109
0.0115
481/481
0.0123
124/481
0.0128
144/144
0.0134
40/144
0.0148
22/22
0.0124
9/22
0.0136
6/6
0.0084
1/6
0.0037
ΤRDS = 100 s
Nr/Nrtot
ξ
344/1838
0.0046
357/2361
0.0125
348/2494
0.0089
282/1907
0.0098
183/1149
0.0085
91/505
0.0078
33/128
0.0105
8/38
0.0077
1/5
0.0028
ΤRDS = 100 s
Nr/Nrtot
ξ
343/1820
0.0090
365/2404
0.0129
363/2500
0.0187
299/2003
0.0117
198/1109
0.0125
96/481
0.0134
34/144
0.0146
9/22
0.0136
1/6
0.0037
*mean of peak values.
50 s cases by again generating more averages. Excessive overlap was not a consideration
for the local extrema condition, unlike its positive point counterpart, as this stricter
triggering condition using only peaks already minimizes the potential for overlap.
One final interesting observation can be made for the high amplitude triggers. At
high amplitudes, the limited number of averages prohibits consistent performance.
However, even with so few averages, these triggers may produce more reasonable
42
TABLE 2.5 (CON’T)
X0
1.55
2.33
3.10
4.66
6.21
7.76
9.32
10.8
12.4
3.74*
ΤRDS = 0 s
Nr/Nrtot
ξ
268/268
0.0206
449/449
0.0160
660/660
0.0103
670/670
0.0111
547/549
0.0112
295/297
0.0105
99/99
0.0103
26/26
0.0084
2/2
0.0064
744/744
0.0108
X0
1.50
2.25
3.00
4.50
6.00
7.50
9.00
10.5
12.0
3.51*
ΤRDS = 0 s
Nr/Nrtot
ξ
280/280
0.0315
519/519
0.0306
702/702
0.0197
790/790
0.0157
546/546
0.0119
300/300
0.0121
102/102
0.0148
20/20
0.0109
5/5
0.0046
740/742
0.0140
Abs. Value Peak Trigger
Run I
ΤRDS = 50 s
Nr/Nrtot
ξ
162/268
0.0140
262/449
0.0151
330/660
0.0123
312/670
0.0122
211/549
0.0114
108/297
0.0085
38/99
0.0086
8/26
0.0096
½
0.0028
273/744
0.0096
Run II
ΤRDS = 50 s
Nr/Nrtot
ξ
181/280
0.0315
283/519
0.0349
350/703
0.0182
344/790
0.0162
241/546
0.0128
118/300
0.0115
39/102
0.0153
8/20
0.0120
1/5
0.0037
346/742
0.0129
ΤRDS = 100 s
Nr/Nrtot
ξ
139/268
0.0043
196/449
0.0153
236/660
0.0132
220/670
0.0119
165/549
0.0126
86/297
0.0079
32/99
0.0089
8/26
0.0096
1/2
0.0028
236/744
0.0085
ΤRDS = 100 s
Nr/Nrtot
ξ
154/280
0.0353
208/519
0.0309
247/703
0.0182
249/790
0.0153
179/546
0.0133
93/300
0.0124
33/102
0.0142
8/20
0.0120
1/5
0.0037
245/742
0.0165
*mean of peak values.
damping estimates than the same number of averages would at a lesser amplitude levels.
Stability is achieved when a high number of averages are present, though this will never
be the case for the amplitude values which lying in the tails of the distribution.
Intuitively, at high amplitudes the points satisfying the trigger condition are
closely associated with maximum values within the signal. They may represent a pulse or
high amplitude burst within the signal. However, following this pulse, before the arrival
43
of the next significant excitation, the amplitude will naturally tend to diminish, leading to
a lull of the time history. As a result, the very decay characteristic, which is being
isolated in the RDT, surfaces more likely at these higher amplitudes than at their lower
counterparts. This may help to explain why higher amplitude triggers from 1-3 σ
outperform other triggers that may afford more averages.
To illustrate, consider Run III for the trigger condition of Xo=11.9 (4σ), shown in
bold in Tables 2.2 and 2.4. The six captured segments meeting the positive point
condition for TRDS = 0, and later averaged to form the RDS, are shown in Figure 2.8. The
dotted line is the theoretical free vibration for this case. If the other two correlation
conditions (TRDS = 50 and 100 s) were to be enforced, two of theses segments, X3(t) and
X4(t), would be negated. On the other hand, the even stricter conditions imposed by the
local extrema condition would select only X3(t), X4(t) and X5(t) for TRDS = 0 and if no
overlap between captured segments is permitted, only X3(t) and X5(t) would be retained.
It should be noted that indeed, at this high amplitude, some of the segments already
closely resemble the decay curve for the system, implying that fewer averages are
required for meaningful results. As opposed to the segments captured at smaller
amplitudes, such as those shown by the top two images at the left of Figure 2.2, where the
captured segments in no way resemble a decay. Unfortunately, there are few segments at
these higher amplitude levels and typically not enough to average the RDS into a fully
stable signature.
44
X2(t)
X6(t)
X4(t)
X1(t)
X3(t)
X5(t)
t [s]
t [s]
FIGURE 2.8. Captured random decrement segments for Run III,
Xo=11.9. Dotted line indicates theoretical free vibration decay at this
amplitude level
2.2.2.2.5 Influence of Nonstationarity on Random Decrement Signatures
As the requirements of lengthy stationary data sets had often precluded the use of
traditional spectral and autocorrelation techniques, the RDT was proposed as a way to
circumvent this problem by permitting analysis on shorter lengths of stationary data.
Though it is often assumed that wind-induced response of structures is stationary,
examination of full-scale data has often demonstrated otherwise. Even so, there has been
little treatment of the ability of RDT to perform under nonstationary conditions (Jeary,
45
1992). To illustrate the implications of stationarity on the RDT, two signals were studied
using the same oscillator defined in Section 2.2.2.2. In signal 1, the random input is
comprised of two, 4-hour blocks of standard Gaussian white noise (termed segments 1 &
3) separated by a 4-hour block of zero mean, white noise drawn from a uniform
distribution (termed segment 2). The definitions of stationarity require that all statistical
properties be invariant with time, strictly implying that all the data be drawn from the
same distribution. A sudden pocket of nonstationarity is introduced to violate this
assumption.
Figure 2.9 illustrates the implications of the violation of stationarity in signal 1 by
examining the bootstrapped variance for several triggering thresholds. Note that in each
of these cases, the same number of segments (Nr=200) was averaged. Although the
variance is theoretically independent of triggering level, as evidenced by Equation 2.17,
this figure displays an increase in variance with triggering level when compared to the
limiting variance function of Equation 2.22. These findings may be rationalized by in
light of the histograms of the peaks within each block of data, which were omitted for
brevity. Over ninety percent of the high amplitude peaks are located in segments 1 & 3.
As a result, when using these higher amplitude trigger conditions, shown in blue, there
may be only a few isolated samples drawn from segment 2. The Gaussian samples by
themselves are incapable of averaging out the variance introduced by the isolated samples
captured from segment 2, thus leading to corrupted results for these higher trigger levels.
Conversely, low amplitude trigger conditions shown in shades of red have a sufficient
number of samples drawn from segment 2 to cancel out the variance from the uniformly
distributed response component. The presence of such localized pockets of
46
FIGURE 2.9. Bootstrap variance estimates for nonstationary signal 1 with error expression in Equation
2.22 shown by solid line (Nr=200)
nonstationarity appears to be the culprit Jeary (1992) hoped to negate by establishing the
conditions for feature removal bulleted in the previous section. Through the use of such
pre-treating in the time history, these local nonstationary features can be removed to
improve the overall reliability of the RDS.
A second nonstationary signal was investigated by enveloping the excitation by a
sinusoidal function. As opposed to the previous instance, this case simulates a global
phenomenon, with a sinusoid of 2-hour period modulating the Gaussian, white noise
excitation. As shown by Figure 2.10, in this case the dependence upon triggering
amplitude is of course not present, as all levels of triggering reflect the same poor
performance when compared to variance estimate in Equation 2.22, although higher
trigger levels seem to scatter toward higher variances. This reflects the power of global
47
FIGURE 2.10. Bootstrap variance estimates for nonstationary
signal 2
nonstationary features in the data, as the variance of estimates will be far greater than
idealized theory would predict.
2.3 Uncertainty in System Identification and Damping Estimation
System identification by any approach represents the process through which optimal
values of system parameters are obtained, though this is complicated without the benefit
of measured input. In this process, the variability in the PSD and RDS is of concern.
While some understanding of uncertainty is available, through Equations 2.10 and 2.17,
specific measures of variance in frequency or damping estimates are in general not
known. As the exercises in Section 2.2.2.2.4 and Figure 2.5 demonstrate, damping values
48
can fluctuate, even when the variance is supposedly minimized. Seybert (1981)
recognized this concern and derived approximate expressions for the bias and random
errors in the damping estimation using the half-power bandwidth of power spectra. Even
though these depend on the very quantity being estimated, they have proven insightful
and indicate that the normalized bias in the damping estimate is approximately 1.5 times
the normalized bias of the power spectrum. This derivation was made realizing that the
bias of the PSD itself in the vicinity of the HPBW points will directly affect the estimates
of damping. Seybert (1981) also derived an approximate expression for the normalized
variance of the damping estimate, again confirming that the damping variance is
proportional to the number of raw spectra considered. However, the expression has
limited utility in this case, as the measure of the coherence function between the input
and output processes at the HPBW points is required. Another option is empirical
expressions for the coefficient of variation, such as those determined by Montpellier
(1996). Though useful, these expressions are again approximate and idealized, an issue
that will be addressed in the following section.
To explore the variability in system identification by SA and RDT, frequency and
damping are estimated from the response of ambiently-excited SDOF oscillators sampled
at 10 Hz. In accordance with common practice, the minimum required spectral resolution
is determined by limiting the normalized spectral bias error in Equation 2.8 to less than
-2% and determining the required number of FFT points (N) conservatively to the nearest
power of two. To minimize normalized variance errors to 10%, enough data is generated
to yield 100 sufficiently resolved raw spectra. These time histories are generated by
exciting a SDOF system with a band limited Gaussian, zero mean white noise process.
49
This random input was simulated from a target band-limited spectrum, with unit
magnitude from 0 Hz to a specified cutoff frequency fc (Li & Kareem, 1993a). The
resulting simulation parameters are documented in Table 2.6 for three cases: Cases A, B
and C.
Though often not obtainable in practice, significant amounts of stationary data are
considered for illustrative purposes. The first case represents a lightly damped 1 Hz
oscillator. The latter two cases involve more narrowband systems, requiring significantly
more data to achieve the same spectral bias and variance errors. In Case C, though the
same amount of data is generated, the length of segments captured in the RDT is
shortened from TRDS = 60 s to TRDS = 30 s in order to increase the number of segments
being averaged, with assumedly minimal influence due to correlation. To demonstrate the
worst-case performance of the RDT, the local extrema trigger condition is enforced,
selecting a value of 3% the median peak value to generate most possible segments.
Individual time histories generated by this approach were retained for the subsequent
resampling analysis presented in Sections 2.4.1 and 2.4.2.
Armed with this data, the normalized bias is assumed to be less than -2% and the
normalized variance to be 10%, but the exact error in the frequency and damping
identified from the PSD is still not known. To illustrate the variability possible, the
systems in Table 2.6 are each simulated 50 times. In each case, the same amount of data
is used so that the normalized bias and random errors on paper are the same, and the
system is then identified by both the spectral analysis and Random Decrement Technique
discussed previously. Further, since the variance in the RDS increases with each cycle of
50
oscillation, system identification is applied over only the first two cycles of the RDS.
Since the logarithmic decrement was shown in Figure 2.5 to manifest variability in this
range, a least squares fit of Equation 2.14 is utilized for damping identification.
TABLE 2.6
PROPERTIES OF SIMULATED CASES
CASE
A
B
C
fc
(Hz)
5
1
1
fn
(Hz)
1.0
0.2
0.2
ξ
N
Ns
Nra
0.01
0.01
0.01
4096
16384
16384
100
100
100
~600-700
~700-800
~1100-1200
Length of Data
(Hr)
11.4
45.5
45.5
a
Actual Nr will very in each run dependent upon the number of peaks forming nonoverlapping segments.
Even in the presence of a favorable amount of data, the variability inherent in this
random process can cause considerable scatter in the identified parameters, as shown by
Table 2.7, summarizing the statistics of the simulations, including the mean, standard
deviation, and coefficient of variation. From Table 2.7 and Figure 2.11, which
graphically displays the data for Case B, it is not surprising to confirm that the frequency
is relatively simple to identify with accuracy. The damping, on the other hand, is much
more difficult. As expected, SA produces a biased overestimation of the damping present,
as the smoothing of the spectrum results in an underestimation of the spectral peak and
thus an overestimation of the damping. The fact that the SA bias is consistent should not
be surprising since the goal was to maintain the same level of bias. Though the
normalized bias in the power spectrum is less than –2%, the normalized bias in the
51
(a)
(a)
(b)
(b)
FIGURE 2.11. Identified dynamic properties of simulated data: identification
of Case B by (a) spectral analysis and (b) Random Decrement Technique
TABLE 2.7
STATISTICS OF MONTE CARLO SIMULATIONS
µ[fn]
(Hz)
σ[fn]
(Hz)
bias [fn]
(Hz)
SA
RDT
1.0023
1.0003
0.0012
0.0014
0.0023
0.0003
SA
RDT
0.2007
0.2001
0.00026
0.000256
0.0007
0.0001
SA
RDT
0.2006
0.2000
0.00027
0.00025
0.00060
-0.00002
CoV [fn]
CASE A
0.12%
0.14%
CASE B
0.13%
0.13%
CASE C
0.13%
0.12%
52
µ[ξ]
σ[ξ]
bias[ξ]
CoV[ξ]
0.01041
0.00921
0.00029
0.00124
0.00041
-0.00079
2.76%
13.50%
0.01056
0.00995
0.00033
0.00126
0.00056
-0.00005
3.12%
12.67%
0.01052
0.01011
0.00032
0.00109
0.00052
0.00011
3.01%
10.79%
damping estimated from it is several times that amount. Recall that Seybert’s (1981)
expression underestimated this over prediction as only 1.5 times the normalized PSD
bias. As expected, the bias does become a bigger problem in the narrowband system. For
Case C, this narrowband system was resimulated, with the same amount of data as in
Case B. As a result, the marginal changes in the statistics associated with the SA are only
the result of the new random excitation generated in simulation. The variance is largely
unchanged with standard error changing only by 4% between the two cases. The CoV for
the SA in all cases is between 2.75 and 3.15%. It should be relatively constant for all
cases as the length of data was selected to maintain the same bias and random errors in
the PSD. Note that the variance estimates in Table 2.7 are still about one-half the
estimates provided in Montpellier (1996).
Though both approaches have near identical performance with respect to
frequency identification, there is, unfortunately, a considerable amount of scatter in the
RDT damping result, leading to a CoV that is an order of magnitude greater than SA
result. This CoV is nearly half that observed by Montpellier (1996), as expected due to
the large amount of data considered. The level of bias in the RDT result is more
significant in the relatively broader band system, perhaps due to the limited number of
segments being averaged. However, when large amounts of segments are available, the
estimate of damping is nearly unbiased, as expected. Note that since the least squares fit
is used, the biased observed in LogDec damping estimates in Section 2.2.2.2.4 is negated.
This is advantageous since narrowband systems are increasingly difficult to identify by
spectral analysis. The larger number of RDS averaged in Case C helps to further reduce
the deviation of damping estimates by RDT. Increasing Nr by a few hundred samples has
53
led to a nearly a 15% decrease in the standard deviation and as shown in Section
2.2.2.2.4, the negative impacts of correlation are often offset by this increase in the
number of averages.
It is important to note that due to the random nature of the process identical
amounts of data subjected to the same SA can produce an estimate of damping which has
no error or as much as 10% error. Though 10% error may be reasonable, this situation
also illustrates a very favorable and perhaps unrealistic amount of stationary data. Under
these ideal conditions, the damping estimated by independent simulations has inherent
variability, which is no doubt enhanced under less ideal conditions. On the other hand,
the RDT identification of the system from any one of these simulations can produce
estimates of damping that are near exact or in error by up to 20-30%. Though Table 2.7
illustrates that if one has the luxury of repeating an experiment 50 times under identical
conditions, acceptable results are obtainable in the average, it is beneficial to be able to
assess the accuracy of identified parameters from a given time history. The accuracy of
any identification by one of these two approaches is contingent upon the level of
randomness in the measured data and the degree to which it has been eliminated in the
averaging process.
2.4 Variance of Damping Estimates via Bootstrapping
Typically, without the advantage of repeated experiments or measured input, the dynamic
analyst is forced to make the best of only limited observations to perform system
identification and determine the reliability of that estimate. Non-parametric resampling
54
schemes such as the bootstrapping technique introduced in this chapter and discussed
further in the Appendix can be useful in such cases. These approaches amount to treating
the observed samples as if they are representative of the larger population, then
resampling this limited data to approximate variance. Using this computationally simple
scheme, the analyst can take the time history that would normally result in a single
estimate of damping with no variance measure and effectively “extend” that data through
resampling in order to approximate the variance of that estimate. Such approaches further
allow the assignment of confidence intervals without necessitating knowledge of the
parameter distribution. This practical tool is introduced here to quantify random errors of
damping estimates garnered from these two common system identification approaches.
It should be noted that other system identification techniques have also considered
the issue of statistical reliability of the identification process. For example,
Autoregressive Moving Average (ARMA) schemes (e.g. Spanos & Zeldin, 1996; Li &
Kareem, 1993b) or simplified Autoregressive (AR) models utilizing the Maximum
Entropy Method, e.g. Cao et al. (1997) implicitly provide performance measures in terms
of prediction errors, which indirectly reflect the quality of the AR or ARMA
identification. More recently, a Bayesian spectral density approach was introduced by
Katafygiotis & Yuen (2001) to update the PDF of modal parameters for ambientlyexcited data. The resulting PDF obtained in the minimization scheme was found to be
well approximated by a Gaussian distribution, whose mean is indicative of the optimal
parameter estimates and covariance yields a direct measure of uncertainty. Approaches
like these provide a variance estimate of the damping parameter as a byproduct of the
minimization operation or “goodness of fit” to an assumed model. The associated errors
55
then indicate how well the estimated damping parameter value fits the available data,
providing one form of reliability measure.
On the other hand, the intent of this work is not to propose an altogether new SI
technique with error estimates to quantify a “goodness of fit.” Rather, this study provides
a practical computational tool to generate the statistical reliability measures for frequency
and damping parameters estimated using traditional approaches like SA and RDT, for
which no statistical confidence measures are implicitly provided. It is the inherent
randomness in the data that influences the quality of the PSD and RDS and thereby the
damping estimates garnered from them. As these schemes must average out the effects of
randomness, their performance becomes dependent upon sufficient amounts of available
data. The intent of this work is to mimic this random characteristic to provide a
supplementary tool to gauge the potential variance in the resulting damping estimate from
a given spectral or random decrement analysis.
For each case in Table 2.6, the bootstrapping scheme outlined in the Appendix is
applied to selected time histories simulated by Monte Carlo in Section 2.3. To illustrate
the information that can be gained, particularly when limited data is available, increments
of the total time history are analyzed so that the influence of the number of raw spectra
(Ns) and the number of segments in RDT (Nr) can be determined. As the frequency is
estimated with near certainty every time, it shall not be discussed here for brevity. The
tables that follow contain the bootstrap estimate of the parameter, in this case damping,
ξboot, defined as the mean of the parameter identified from B=1000 bootstrap replicates
(see Appendix Equation A.2). The traditional estimate of the parameter, without any
56
resampling, is commonly termed the plug-in estimate, ξplugin. The standard deviation of
the bootstrap replicates σΒ, as defined by Equation A.1, and the CoV, defined as the ratio
of the standard error to the bootstrap mean in Equation A.2, are also provided. The
bootstrap bias in the estimate is then defined as the difference, ξboot - ξplugin. This measure
is important, as a small bootstrap bias, relative to the standard error, confirms that the
bootstrap is a good estimate of the parameter, e.g. bias/σ < 0.25. As this measure
becomes larger, the bootstrap estimates may no longer be accurate and require bias
correction, which is discussed further in Efron & Tibshirani (1993). Additionally, by
virtue of the bootstrapping scheme, histograms depicting the distribution of the damping
estimate from a given time history can be obtained. These are useful tools for identifying
the underlying distribution of damping estimates and its associated characteristics. An
example of such a histogram for the damping estimates from a single time history (Case
B) is given in Figure 2.12. While viewing these results, please be reminded that the
simulated system had a critical damping ratio of 0.01.
Additionally, confidence intervals are defined for the damping estimate. By
traditional analysis, only a single estimate of damping is available from each simulated
time history. However, through the bootstrapping scheme, this estimate is enhanced by a
family of associated estimates, which can give valuable insight into the reliability of a
given damping estimate. In the most elementary formulation, a level of confidence can be
selected, and then the replicates that correspond to this level can be identified from the
resampled distribution (Efron & Tibshirani, 1993). By virtue of the non-parametric nature
of this approach, there is no need to make any assumptions about the Normality of the
57
Ns=10
Ns=20
Ns=30
Ns=40
Ns=50
Ns=70
Number of occurrences
Number of occurrences
(a)
Ns=90
Ns=60
Ns=80
Ns=100
Damping Estimate [%]
Damping Estimate [%]
Nr=100
Nr=200
Nr=300
Nr=400
Nr=500
Number of occurrences
Number of occurrences
(b)
Nr=700
Nr=600
Nr=800
Damping Estimate [%]
Damping Estimate [%]
FIGURE 2.12. Histogram of bootstrap simulations on (a) PSD and
(b) RDS for Case B
58
damping estimates. This is especially useful in cases where the PSD or RDS is generated
from a limited number of samples. However, when the CoV of the bootstrapped
replicates is small, e.g. less than 20% and the bootstrap biases are negligible, a Gaussian
PSD may be assumed to describe the distribution of the bootstrapped damping parameter.
The 95% confidence bands are then approximated to lie within 2 standard deviations of
the bootstrap mean (Efron & Tibshirani, 1993). These bands are provided in the last
column of the following tables for comparison.
2.4.1 Discussion: Bootstrapping Damping Estimates for Spectral Analysis
As shown in Table 2.8, the bootstrap investigations conducted on a single record from the
Monte Carlo simulations in Section 2.3 predict significantly smaller values of standard
errors than anticipated for spectral analysis. However, the relative changes are consistent
with Monte Carlo predictions, when examining the full data set (Ns=100), shaded in gray.
Case A has smallest variance, thus one would expect the Case A bootstraps to reflect that
smaller variance in comparison to Cases B and C. Likewise, Case C produced a slightly
smaller value of standard error than Case B, a trend reflected in the bootstrap errors.
From the sample histogram in Figure 2.12, it is evident that in cases of limited data there
are shifts in the damping estimate distribution about various mean values. It is only for
Ns>50 that the behavior stabilizes and the damping estimates distribute about a relatively
constant mean, indicating that the variance is sufficiently minimal.
It is interesting to note that, even with modest amounts of data (e.g. Ns=10), a
relatively good estimate of the damping is obtainable, while the addition of further
59
TABLE 2.8
STATISTICS OF BOOTSTRAP REPLICATIONS OF CRITICAL DAMPING
RATIO: ESTIMATES BY SPECTRAL ANALYSIS
CASE A
bias[ξ]
σ[ξ]
Ns
ξboot
ξplugin
10
0.010299
0.01030
-0.000011
0.000084
0.82%
20
0.010478
0.010481
-0.000029
0.000082
0.78%
30
0.010571
0.010564
0.000064
0.000081
0.77%
40
0.010167
0.010165
0.000016
0.000079
0.78%
50
0.010099
0.010096
0.000032
0.000078
0.77%
60
0.009940
0.009940
0.000041
0.000080
0.80%
70
0.010127
0.010131
-0.000037
0.000082
0.81%
80
0.010075
0.010079
-0.000044
0.000079
0.78%
90
0.010043
0.010041
0.000019
0.000076
0.76%
100
0.010029
0.010027
0.000013
0.000083
0.83%
CoV
95%
bootstrap
(0.01016,
0.01044)
(0.01034,
0.01060)
(0.01044,
0.01071)
(0.01004,
0.01030)
(0.00997,
0.01023)
(0.00981,
0.01007)
(0.00999,
0.01025)
(0.00995,
0.01020)
(0.00991,
0.01017)
(0.00989,
0.01016)
95%
normal
(0.01013,
0.01047)
(0.01031,
0.01064)
(0.01041,
0.01073)
(0.01001,
0.01032)
(0.00994,
0.01025)
(0.00978,
0.01010)
(0.00996,
0.01029)
(0.00993,
0.01023)
(0.00989,
0.01019)
(0.00986,
0.01019)
95%
bootstrap
(0.01036,
0.01052)
(0.01053,
0.01101)
(0.01088,
0.01130)
(0.01041,
0.01077)
(0.01027,
0.01059)
(0.00996,
0.01029)
(0.01048,
0.01088)
(0.01043,
0.01086)
(0.01043,
0.01089)
(0.01039,
0.01083)
95%
normal
(0.01034,
0.01054)
(0.01049,
0.01106)
(0.01084,
0.01136)
(0.01037,
0.01079)
(0.01024,
0.01062)
(0.00991,
0.01033)
(0.01043,
0.01091)
(0.01038,
0.01091)
(0.01041,
0.01093)
(0.01035,
0.01087)
CASE B
Ns
ξboot
ξplugin
bias[ξ]
σ[ξ]
CoV
10
0.01044
0.01044
-0.000001
0.000049
0.47%
20
0.01077
0.01080
-0.000031
0.000143
1.33%
30
0.01110
0.01110
-0.000005
0.000130
1.17%
40
0.01058
0.01061
-0.000033
0.000105
0.99%
50
0.01043
0.01044
-0.000015
0.000095
0.91%
60
0.01012
0.01012
0.000002
0.000104
1.03%
70
0.01067
0.01075
-0.000081
0.000121
1.13%
80
0.01064
0.01065
-0.000009
0.000134
1.26%
90
0.01067
0.01069
-0.000024
0.000130
1.22%
100
0.01061
0.01065
-0.000045
0.000132
1.24%
60
TABLE 2.8 (CON’T)
CASE C
bias[ξ]
σ[ξ]
Ns
ξboot
ξplugin
10
0.010637
0.010637
-0.000001
0.000065
0.61%
20
0.010104
0.010104
0.000000
0.000117
1.16%
30
0.010187
0.010193
-0.000007
0.000113
1.11%
40
0.010453
0.010464
-0.000010
0.000107
1.02%
50
0.010516
0.010522
-0.000006
0.000103
0.98%
60
0.010346
0.010348
-0.000002
0.000122
1.18%
70
0.010089
0.010094
-0.000005
0.000109
1.08%
80
0.009837
0.009840
-0.000003
0.000123
1.25%
90
0.009922
0.009917
0.000005
0.000121
1.22%
100
0.009993
0.009991
0.000002
0.000116
1.16%
CoV
95%
bootstrap
(0.01053,
0.01075)
(0.00992,
0.01030)
(0.01001,
0.01038)
(0.01026,
0.01062)
(0.01032,
0.01068)
(0.01015,
0.01052)
(0.00991,
0.01026)
(0.00963,
0.01005)
(0.00973,
0.01012)
(0.00980,
0.01020)
95%
normal
(0.01051,
0.01077)
(0.00987,
0.01034)
(0.00996,
0.01041)
(0.01024,
0.01067)
(0.01031,
0.01072)
(0.01010,
0.01059)
(0.00987,
0.01031)
(0.00959,
0.01008)
(0.00968,
0.01016)
(0.00976,
0.01022)
recorded data may lead to poorer estimates. This may seem counterintuitive, as the
variance of the PSD has been shown to reduce with the number of raw spectra being
averaged. However, the inherent randomness of the process must be considered. The first
few hours of data may not have significant randomness in comparison to later
components of the overall time history, luckily leading to a reasonable estimate of
damping. Likewise the variance is also dependent on the magnitude of the PSD itself,
which also fluctuates in each case considered. These random fluctuations can and should
be expected when limited data is used, as the variance is high (Vandermeulen et al.,
2000). It is only in the limit that a more stable and reliable PSD results – one without
61
marked fluctuations as more data is considered in the average. As evidenced by Figure
2.12 and Table 2.8, it would indicate that this is achieved for Ns>60 for the narrowband
system, but achieved much sooner, Ns>40 for Case A.
As evident from Figure 2.11, there is a discernable amount of bias in the spectral
estimate of the narrowband system, shifting the mean critical damping ratios to
approximately 0.0105. The inherent bias in the estimate cannot be overcome by the
bootstrapping approach, as also noted in Vandermeulen et al. (2000). Spectra with an
outright bias cannot be enhanced by this approach, as they are not truly representative of
the system to be identified, but rather a biased representation of that system. As the
bootstrap cannot repair sampled data, but can merely make inferences about its various
statistics, the confidence intervals and all relevant statistical distributions will be
clustered about this biased mean, as illustrated by the histograms shown in Figure 2.12.
In this case, even placing 95% confidence intervals on the estimate will not capture the
true damping value. Note that Seybert’s (1981) derivation can provide a normalized bias
estimate for ξ. Knowing that the SA consistently overestimates damping, the rough
normalized bias formula may be offered as a correction to the estimates and used to
refine the confidence bands.
2.4.2 Discussion: Bootstrapping Damping Estimates for Random Decrement Technique
As shown in Figure 2.11, the distribution of RDT damping estimates on the same data
manifest considerably more scatter than SA estimates, consistent with the findings of the
Monte Carlo simulation. The sensitivity of RDT damping estimates was fully discussed
62
in 2.2.2.2.4. It should be noted that an enhancement in performance may be achieved
using the more relaxed positive point trigger or by using the averaging of damping
estimates of adjacent amplitude trigger levels as shown in Table 2.3 to mitigate the
influence of outlier segments. Once again, the distributions in Figure 2.12 shift as more
samples are considered. As shown by Table 2.9, as Nr increases, their behavior tends to
stabilize about a consistent mean value, similar to what was found in SA approach. Even
in this stable range, there is still some fluctuation in the estimate, consistent with the
findings in Section 2.2.2.2.4 requiring the aforementioned local averaging over adjacent
triggers.
However, unlike SA results, the behavior of RDT estimate using local extrema
trigger clearly indicates that limited amounts of data offer little hope of an accurate result,
as many eligible triggers are eliminated by this strict condition. In such cases, the positive
point trigger with TRDS = 0 should be used, as advocated in Section 2.2.2.2.4. Rather, the
results steadily improve with the number of samples being considered: for Nr>500, the
damping estimates are consistently within 10% of the actual value. The level of variance
in the RDT estimate, as shown by Monte Carlo simulation, is markedly greater than the
SA, due again to the sensitivity of trigger amplitude. It can be implied that the SA
approach is more of an averaged approach that does not allow referencing to a trigger
level. Likewise, by selecting a local range of trigger amplitudes in the RDT, as opposed
to just the median peak value used here, the average damping value should be
comparable to those determined by SA, while still preserving the ability to amplitude
reference a damping estimate.
63
TABLE 2.9
STATISTICS OF BOOTSTRAP REPLICATIONS OF CRITICAL DAMPING RATIO:
ESTIMATES BY RDT
CASE A
bias[ξ]
σ[ξ]
Nr
ξboot
ξplugin
100
0.010636
0.010586
0.000051
0.001105
10.39%
200
0.012293
0.012320
-0.000023
0.001144
9.31%
300
0.012927
0.012932
-0.000006
0.001207
9.34%
400
0.011308
0.011340
-0.000032
0.001225
10.83%
500
0.011149
0.011163
-0.000013
0.001264
11.34%
600
0.010915
0.010971
-0.000056
0.001237
11.33%
Nr
ξboot
ξplugin
100
0.01533
0.01532
0.000004
0.001239
8.08%
200
0.01386
0.01381
0.000054
0.001223
8.82%
300
0.01296
0.01296
0.000002
0.001259
9.72%
400
0.01203
0.01188
0.000159
0.001174
9.76%
500
0.01132
0.01126
0.000056
0.001231
10.88%
600
0.00974
0.00976
-0.000013
0.001243
12.76%
700
0.00917
0.00917
0.000005
0.001220
13.30%
800
0.01005
0.01000
0.000048
0.001218
12.13%
CASE B
bias[ξ]
σ[ξ]
64
CoV
CoV
95%
bootstrap
(0.00876,
0.01244)
(0.01041,
0.01417)
(0.01088,
0.01495)
(0.00934,
0.01339)
(0.00901,
0.01323)
(0.00881,
0.01295)
95%
normal
(0.00843,
0.01285)
(0.01001,
0.01458)
(0.01051,
0.01534)
(0.00886,
0.01376)
(0.00862,
0.01368)
(0.00844,
0.01339)
95%
bootstrap
(0.01334,
0.01747)
(0.01181,
0.01591)
(0.01089,
0.01500)
(0.01010,
0.01397)
(0.00921,
0.01333)
(0.00772,
0.01093)
(0.00717,
0.01203)
(0.00808,
0.01201)
95%
normal
(0.01285,
0.01781)
(0.01142,
0.01631)
(0.01044,
0.01548)
(0.00969,
0.01438)
(0.00886,
0.01378)
(0.00726,
0.01223)
(0.00673,
0.01161)
(0.00761,
0.01248)
TABLE 2.9 (CON’T)
CASE C
bias[ξ]
σ[ξ]
Nr
ξboot
ξplugin
100
0.004597
0.004578
0.000019
0.001132
24.61%
200
0.006960
0.007009
-0.000049
0.001151
16.53%
300
0.008277
0.008304
-0.000027
0.001175
14.20%
400
0.008907
0.008922
-0.000015
0.001118
12.55%
500
0.008850
0.008850
0.000000
0.001128
12.74%
600
0.009955
0.009860
0.000095
0.001152
11.57%
700
0.010079
0.010108
-0.000029
0.001189
11.80%
800
0.009909
0.009971
-0.000062
0.001204
12.15%
900
0.010256
0.010229
0.000027
0.001162
11.33%
CoV
95%
bootstrap
(0.00281,
0.00644)
(0.00510,
0.00889)
(0.00635,
0.01014)
(0.00709,
0.01070)
(0.00701,
0.01067)
(0.00812,
0.01196)
(0.00810,
0.01204)
(0.00808,
0.01198)
(0.00834,
0.01219)
95%
normal
(0.00233,
0.00686)
(0.00466,
0.00926)
(0.00593,
0.01063)
(0.00667,
0.01114)
(0.00659,
0.01110)
(0.00765,
0.01226)
(0.00770,
0.01246)
(0.00750,
0.01232)
(0.00793,
0.01258)
This large standard deviation is accurately represented by the bootstrap analysis,
in part due to the fact that the RDT is an unbiased estimator. This was not possible in the
biased SA where the predicted variance was much less than the observed Monte Carlo
result. As the RDT tends to stabilize after a significant number of averages, the changes
in standard deviation are quite small when additional samples are added. Between Cases
A and B, the Monte Carlo standard deviation changes only marginally, by about 2%,
explaining why the bootstrapped estimates for these two cases show only slight
difference. In the same way, the standard deviation takes on consistently smaller values
in Case C than in Case B, reflecting the reduction in variance due to the inclusion of
additional segments in the average. Unlike the biased SA approach, the broad confidence
intervals predicted by bootstrapping RDT results will encase the predicted result at
65
minimum for Nr>400. It is important to emphasize that traditional RDT damping
estimates lack any type of interval estimate. Thus such tools can offer insight where
previously there was none.
Interestingly, a cursory examination of the ratio of bias to standard deviation for
both RDT and SA data confirms that SA generally produces relatively large biases in the
bootstrap data, whose ratios to the standard error exceed 0.25. On the other hand, RDT
produces very small biases in the bootstrap analysis. Large biases often indicate that the
bootstrap analysis of the data may not be accurate, as it was shown that the bootstrap
approach could not capture the variance predicted by Monte Carlo for SA. Bias corrected
data may be resampled, but is should be noted that this is risky practice and is thus not
advocated in this study as it often leads to an increase in standard error. More details may
be found in Efron & Tibshirani (1993).
2.5 The Challenges of Nonlinearity and Nonstationarity
Spectral analysis is marred by the unavoidable presence of nonstationarity concerns that
are inherent in long records. As a result, curve fitting has been advocated to smooth
spectral estimates garnered from shorter but assuredly stationary blocks of data. For
example, Breukelman et al. (1993), Montpellier et al. (1998), and Jones & Spartz (1990)
fit the spectral estimates to a transfer function of a SDOF system by employing a
maximum likelihood estimator or in the latter case, a least squares approach. By selecting
a sample length that is long enough to ensure sufficient frequency resolution, while
avoiding nonstationarity problems, and ensemble averaging records obtained under
66
similar conditions to reduce the variance of spectral estimates, smooth spectra can be
obtained for linear systems. However, the characteristic features of nonlinear systems that
have transfer functions that depart from linear behavior and the response of systems to
nonlinear loading may be obscured in such a curve fitting approach. Though this spectral
representation can provide an “averaged” sense of the structural properties, the inability
to track specific features and properties of a system in time limits the information which
can be gained from the data. The transform’s inability to capture any local features of the
signal results form the fact that it decomposes the signal into a linear combination of
infinite trigonometric functions, which are incapable of recognizing time-varying system
characteristics.
These nonlinearities in either the loading or structural system induce an
amplitude-dependence in the frequency and damping parameters, which further
complicates estimation. In particular, the level of equivalent viscous damping has been
shown to increase considerably with the level of response in the structure (e.g. Jeary &
Ellis, 1981; Ellis, 1996; Fukuwa et al., 1996). This can be explained when considering
again that viscous damping is generally proposed in the equation of motion to represent
all of the various damping mechanisms, resulting in a model that is non-ideal over a wide
range of amplitudes for non-linear systems. As a result, real structures with nonlinear
characteristics, whose damping is identified assuming such a model, will manifest nonlinearity or a damping that changes with amplitude (Ellis, 1998), implying that the system
is then assumed equivalently linear at a given amplitude level.
67
Based on such full-scale observations, it was proposed that damping in structures
be classified into three distinct plateaus (Jeary, 1986) as illustrated by Figure 2.13.
Considering that the boundary damping attributed to the frictional losses between
interfaces of materials contributes significantly to the energy dissipation in structures, the
relative displacement of these interfaces will dictate the amount of friction losses that
occur. The intuitive arguments for this model are as follows: at low levels of motion,
most cracks and joints within the structure are not significantly slipping. Once a sufficient
number of these interfaces are activated, they will dissipate energy in proportion to their
relative displacements, accounting for the linearly increasing portion of Figure 2.13. At
the high amplitude plateau, all interfaces have been activated, thus defining the maximum
level of damping that should not be exceeded unless damage were to occur within the
structure. Note that the slippage of these interfaces can also be assumed to diminish
Total Damping, ξ
stiffness, thereby explaining the reduction in natural frequency with increasing amplitude
Low
Amp
Plateau
Transition
Region
High
Amp
Plateau
Displacement of Roof
FIGURE 2.13. Amplitude-dependent
damping model
68
observed in full-scale data.
An empirical model for this type of behavior was proposed by Jeary (1986) based
upon full-scale data generated using controlled sinusoidal excitations. This model for the
total damping is given by
ξ = ξo + ξ I
XH
H
(2.24)
for a given mode in the structure. ξo is the low amplitude level of damping, ξI is the
incremental damping in the transition region, H is the height of the structure, and XH is
the displacement at the top of the structure. Specific expressions for each variable can be
found in Jeary (1986). Other amplitude-dependent damping expressions were proposed
by ESDU (1983), Lagomarsino (1993) and Tamura et al. (1995).
In light of this
amplitude dependence, any system identification approach should be capable of
accounting for these characteristics, as observed levels of damping must be referenced to
the amplitude of motion.
As a result, the favorable performance of spectral analysis in comparison to the
Random Decrement Technique with local extrema trigger shown in the idealized
demonstrations in Section 2.4 are often negated in real-world applications where
nonlinearities and nonstationarities are present, as the Random Decrement Technique is
capable of detecting amplitude-dependent damping (Jeary, 1992; Tamura & Suganuma,
1996). As shown in Kareem & Gurley (1996) through the use of a nonlinear softening
oscillator, the damping associated with the trigger condition amplitude is estimated
accurately within the first few cycles of the RDS decay, as they are closest to the
69
enforcement of the initial conditions. As a result of the nonlinearity, the damping values
will begin to deviate in later cycles moving further from the trigger condition. As
similarly observed by Tamura & Suganuma (1996), by treating the non-linear system as
locally linear at the triggering condition, the RDS in the first cycle are consistent with the
free-vibration curve of the linear system for the level of damping and natural frequency
associated with the amplitude level specified by the triggering condition. Thus, the use of
RDT for these types of systems should restrict amplitude-referenced frequency and
damping estimates to the values obtained from the first few cycles of oscillation, where
the system is locally linear. Further, when using a selective pre-treatment and processing
of the time history introduced in Section 2.2.2.2.4, time histories with certain localized
non-stationary features can still be processed by the Random Decrement Technique, as
discussed in Jeary (1992).
2.6 Conclusions
This chapter introduced the significance of damping for minimizing wind-induced
response and the challenges associated with its estimation from ambient vibration data.
Two common approaches for system identification were then presented: the Random
Decrement Technique and traditional spectral analysis. The uncertainty associated with
the two approaches was investigated via Monte Carlo simulation, and bootstrapping was
proposed to provide a practical tool for assessing the uncertainty associated with an
estimate of damping by either technique. It was shown that the bootstrap standard
deviation of the RDT damping estimate was consistent the variance of the damping
estimates determined from repeated simulation of the system by Monte Carlo. On the
70
other hand, the bootstrapped standard deviation of the damping estimate derived from
spectral analysis was considerably less than its Monte Carlo counterpart, attributed to the
fact that it was tainted by the bias inherent in the sample population.
The chapter also went into a detailed analysis of the factors contributing to the
performance of the Random Decrement Technique. Although other studies had
considered the ramifications of violating the white noise assumption or the nonlinearity
of the oscillator, none had systematically addressed other issues governing the
performance of the technique. This chapter responded to this deficiency by investigating
the influence of potential correlation between captured segments, the required number of
captured segments to generate a stable signature, the significance of the selected
triggering condition, and the influence of nonstationarity. For the positive point trigger, it
was found that the increased number of segments averaged in the RDS offsets any
negative consequence resulting from allowing more potential correlation. The damping
under these conditions appears to be consistently estimated in most cases when the
trigger amplitude satisfies the following condition: 3σ > Xo > 1σ. Further, when using
the local extrema trigger, it was similarly found that the trigger amplitude corresponding
to this range performed best, as opposed to low amplitude triggers that afforded more
segments for averaging. Regardless, the performance of the RDT stabilizes under any
trigger condition as considerably more segments are averaged, implying that when
limited amounts of data are available, the restrictiveness of the trigger should be relaxed
via the positive point condition. Finally, though the Random Decrement Technique was
found to accommodate mild nonlinearity and localized nonstationarity, the prevalence of
these features requires a modified analysis framework.
71
2.7 Moving into the Time-Frequency Domain
As nonlinearity and nonstationarity, more often that not, characterize the processes of
interest in Civil engineering, there was a pressing need to develop analysis tools that
could accommodate these features. Although the Fourier Transform revolutionized signal
processing and its applications to various disciplines perhaps like no other development,
its inability to adequately handle nonstationary and nonlinear phenomenon has proven
problematic. Even the Random Decrement Technique’s ability to provide nonlinear
analysis and accommodate nonstationarity to some degree is a limited victory in light of
the amount of data required to successfully obtain the decrement signature. The
inadequacy of such techniques rooted in stationary assumptions becomes even more
apparent when signals with significant nonlinearities or localized frequency
characteristics are present. Recognizing the shortcomings of these more traditional
approaches, a thrust towards new analysis frameworks such as time-frequency analysis is
unfolding, as discussed in Chapter 3.
72
CHAPTER 3
TRANSITION TO THE TIME-FREQUENCY DOMAIN
3.1 The Departure from Fourier Analysis
While the contributions of the Fourier Transform to signal processing cannot be denied,
the fact that actual data is often characterized by localized features obscured by the
infinite bases of the Fourier Transform prompted a departure from this classical
approach. To demonstrate, consider the waveform in Figure 3.1. The softening of
frequency over an isolated time interval is not reflected in the resulting power spectrum.
Even if the Fourier spectrum had detected this contribution and represented it as a
secondary peak, there would be no indication of the duration of this component within
the signal, as all time information is discarded in the transform.
In these cases, it becomes necessary to move to another analysis domain governed
by transforms that produce a time-frequency distribution (TFD) of energy. The
introduction of these transforms can be traced back to the mid-1940’s, based upon the
work of Wigner (1939) in quantum mechanics. The applications of these concepts in
signal processing is largely credited to the work by Ville (1948) and Gabor (1946) and
73
x(t)
Time [s]
Frequency [Hz]
FIGURE 3.1. Sine wave with local frequency change (top) and
traditional power spectral magnitude
are discussed in further detail in Carmona et al. (1998) and Mallat (1998). In this chapter,
the theory developed by various sources are unified and organized, and the details
relevant to Civil Engineering are highlighted. As the essential components of wavelet and
time-frequency analysis theory are introduced, anecdotes and subsequent discussions are
provided to assimilate the various components of the theory, largely grounded in
mathematics and signal processing, and recast it in the unique perspective of Civil
Engineering applications.
74
3.2 Short-Time Fourier Transform
To overcome the limitations of Fourier Transforms, a Short-Time Fourier Transform
(STFT) was developed by Gabor (1946). Since the harmonic waves used in traditional
Fourier analysis are continuous, it is impossible for this representation to have a local
spectral density suitable for uncovering transient information. Thus, the Fourier
Transform was modified by a short-duration window w(t) centered at time, τ. The
spectral coefficients could then be determined over this short length of data, which is
assumed to be stationary, yielding the STFT
STFT (τ , f ) =
∞
∫ x(t )w (t − τ )e
*
−i 2πft
dt
(3.1)
−∞
where * denotes complex conjugate and i is the imaginary number. This new breed of
transform was one of the first time-frequency distributions, named for their ability to
depict energy densities as a dual function of frequency and time. The transform,
appropriately nicknamed the Gabor Transform, became quite popular due to its intuitive
appeal in light of the widely used Fourier Transform, conserving energy between the time
and frequency domain in a similar manner.
The performance of this transform is greatly dependent upon the choice of
window function, which can simply be the traditional boxcar, i.e. rectangular, window.
Proper windows are those with good localization in both the time and frequency domains
and are chosen subjectively based on the application. Other popular choices include the
Hanning and Hamming windows and the Gaussian function discussed in Section 3.2.2
(Carmona et al., 1998). An energy representation, commonly known as the spectrogram,
75
can then be obtained by plotting the squared modulus of the transform to highlight the
time-frequency distributions of energy in the signal. While the notion of a moving
Fourier Transform is attractive for uncovering localized signal features, it still is plagued
by the same issues facing the traditional Fourier Transform.
3.2.1 Heisenberg Uncertainty Principle
As discussed in Chapter 2, spectral analysis is limited by the fact that high resolution
cannot be obtained in both the time and frequency domains as a result of their inverse
relationship. Contracting a function in time to improve its localization will dilate it in the
frequency domain. The measure of time duration, Τx, and frequency bandwidth, Βx, of a
signal has been defined in many ways, but the most common definition was proposed by
Gabor (1946):
∞
Tx2 =
∫t
2
2
x(t ) dt
−∞
∞
(3.2)
∫ x(t )
2
dt
−∞
∞
B x2 =
∫f
2
2
Xˆ ( f ) df
−∞
(3.3)
∞
∫
2
Xˆ ( f ) df
−∞
where x(t) is a zero mean signal and Xˆ ( f ) is its Fourier Transform. These in essence
serve as a measure of the mean square value of the signal in both time and frequency
normalized by the variance or total energy content. The product of this bandwidth and
time duration, often termed BT product, is constant, with its minimum value possible
obtained for a Gaussian signal,
76
B x Tx ≥
1
.
4π
(3.4)
This relation is commonly termed the Heisenberg Inequality or Heisenberg Uncertainty
Principle. This important development defines the primary constraint on time-dependent
spectra, as one cannot obtain a high degree of resolution in both the time and frequency
domains simultaneously. A rigorous proof of this principle may be found in Chui (1992).
Similarly, this definition may be used to define the resolution of a particular zero
mean window function:
∞
∆t w =
2
∫t
2
2
w(t ) dt
−∞
∞
∫ w(t )
(3.5)
2
dt
−∞
∞
∆f w =
2
∫
2
f 2 Wˆ ( f ) df
−∞
∞
∫ Wˆ ( f )
(3.6)
2
df
−∞
where w(t) is the window function and Wˆ ( f ) is its Fourier Transform. The relation in
Equation 3.4 still applies and the product of these two resolutions must be greater than or
equal to 1/4π. As the Gaussian functions establish the lower bound in Equation 3.4, they
are typically chosen as the window function w(t) in the STFT. Note that analogs to
Equations 3.3-3.6 for windows or signals that are not zero mean are provided in Carmona
et al. (1998).
77
3.2.2 Resolutions of the STFT: The Heisenberg Box
Based on the frequency and time resolutions in Equations 3.5 and 3.6, the STFT at time t
and frequency f represents the signal content in a distinct area of the time-frequency plane
defined by the Heisenberg box with the following dimensions
[t − ∆t w , t + ∆t w ]× [ f
− ∆f w , f + ∆f w ] .
(3.7)
The dimensions of the box do not change with the frequency being analyzed but remain
fixed: the box having a width of 2∆fw along the frequency axis and 2∆tw along the time
axis. As argued by Chui (1992), since frequency is directly proportional to number of
cycles in a specific interval of time, a narrow time window is required to locate highfrequency phenomena and a wide time-window is necessary for more thorough
investigation of low-frequency phenomena. As a result, this transform is not well suited
for analysis of signals that may have both low and high frequency components. This is
demonstrated schematically in Figure 3.2, as fewer cycles of the low frequency
component of the signal are considered in the fixed Heisenberg box of the STFT.
Similarly, the time resolution does not become finer at high frequencies, where it is
required to separate rapidly varying high frequency components. To demonstrate,
consider the sum of two hyperbolic chirps shown in Figure 3.3. The STFT’s spectrogram
cannot track the rapidly varying frequencies of the two chirps due to its non-optimal
resolutions, resulting in a blurring of their energy in the high frequency range.
78
f
2∆fw
f2
2∆tw
2∆tw
2∆fw
f1
t
t1
t2
FIGURE 3.2. Concept of Heisenberg box for short-time Fourier
Transforms
3.3 Moving Toward the Wavelet Domain
It becomes evident that by breaking the signal down into a series of basis functions of
finite length, one can overcome some of the limitation of stationarity. As Fourier basis
functions are localized in frequency but not in time, an alternative was introduced in the
previous section to localize the Fourier Transform through the addition of a window
function. However, constraints of the Heisenberg Uncertainty Principle limit the
obtainable resolutions and render them non-optimal over a range of frequencies. An
alternative approach would be to design basis functions that are local in both frequency
and in time and then scaled via dilations to optimally adjust their resolutions based on the
frequency being analyzed, yielding a multi-resolution analysis.
79
x(t)
f [Hz]
t [s]
FIGURE 3.3. Sum of two hyperbolic chirps and
resulting spectrogram (Mallat, 1998)
The Wavelet Transform (WT) was engineered with this in mind. Through an
adaptation to the Heisenberg Uncertainty Principle, the Wavelet Transform concedes that
arbitrarily good resolution in both time and frequency is not possible. Thus the transform
optimizes its resolutions as needed. It provides good resolution at high dilations or low
frequencies, sacrificing its time resolution in this regime to satisfy the Heisenberg
Uncertainty Principle. As the variations of low frequency components with time are
typically gradual, this loss is not of consequence. In the time domain, the transform has
good resolution at high frequencies in order to identify transient signal features. The
fundamental differences between the Fourier Transform, the Short Time Fourier
Transform and the Wavelet Transform are shown in Figure 3.4. The advantages of the
Wavelet Transform approach are evident upon revisiting the two hyperbolic chirps in
Figure 3.3. A plot of the energy distribution from the Wavelet Transform of the signal is
80
WAVELET
TRANSFORM
Frequency
STFT/GABOR
TRANSFORM
Frequency
Frequency
FOURIER
TRANSFORM
Time
Time
FIGURE 3.4. Evolution of time-frequency analysis
provided in Figure 3.5 and demonstrates that the multi-resolution time-frequency analysis
can identify the two components.
3.4 The Transform
The relations governing Wavelet Transforms have been articulated and proven in a
number of texts, e.g. Chui (1992), Mallat (1998) and Carmona et al. (1998). Therefore,
only relevant details are presented herein for brevity. The continuous Wavelet Transform
(CWT) is a linear transform that decomposes a signal x(t) via basis functions that are
simply dilations and translations of the parent wavelet g(t) through the convolution with
the signal according to
W ( a, t ) =
1
a
∞
⎛τ − t ⎞
⎟dt
a ⎠
∫ x(τ )g ⎜⎝
*
−∞
(3.8)
where * denotes complex conjugate. Dilation by the scale, a, inversely proportional to
frequency, represents the periodic or harmonic nature of the signal. The resulting wavelet
coefficients, W(a,t), represent a measure of the similitude between the dilated/shifted
81
f [Hz]
t [s]
FIGURE 3.5. Wavelet transform of two hyperbolic
chirps shown in Figure 3.3 (Mallat, 1998)
parent wavelet and the signal at time t and scale (frequency) a. The normalization by the
root of scale insures that the integral energy given by the wavelet is independent of the
dilation.
Calculation of the Wavelet Transform can be expedited in the frequency domain
through the use of Fourier Transforms. Thus, the convolution in Equation 3.8 can
equivalently be represented as a product in the Fourier domain with Fourier frequency f
∞
W (a, t ) = a ∫ Xˆ ( f )Gˆ * (af )ei 2πft df .
(3.9)
−∞
where X̂ and Ĝ denote the Fourier Transforms of the signal and the parent wavelet.
3.4.1 Completeness
For well-defined parent wavelets (the conditions surrounding this will be discussed in
Section 3.4.2), the inverse transform is given by
82
x (t ) =
1
Cg
∞ ∞
∫ ∫ W (a, t )
−∞−∞
1 * ⎛ t − τ ⎞ da
g ⎜
⎟ 2 dτ
a ⎝ a ⎠a
(3.10)
where Cg is defined as
∞
Cg =
∫
G( f )
2
f
−∞
df .
(3.11)
From the Wavelet Transform, an energy distribution plot called a scalogram can
be generated by plotting the squared modulus of the transform. This representation,
analogous to the spectrogram, represents the energy content of the signal at distinct time
and frequency (scale) pairs
2
SG ( a, t ) = W ( a, t )
2
⎛f ⎞
or SG ( f , t ) = W ⎜⎜ o , t ⎟⎟ .
⎝ f ⎠
(3.12 a, b)
Unlike its STFT counterpart, the energy distribution in the scalogram is multi-resolution.
As demonstrated by proof in Chui (1992) and Mallat (1998), the Wavelet
Transform conserves mean square energy for real signals
Ex = ∫
1
x (t ) dt =
Cg
2
∞ ∞
da
∫ ∫ SG (a, t ) a
2
dt
(3.13)
− ∞−∞
or equivalently
Ex = ∫
1
x(t ) dt = −
Cg
2
∞ ∞
⎛f
∫ ∫ SG⎜⎝
−∞−∞
, t ⎞dfdt
f o ⎟⎠
(3.14)
providing an analog to Plancherel’s Theorem. Thus the wavelet can be viewed either as
measure of time-scale or time-frequency energy density.
83
3.4.2 Conditions on Parent Wavelets
The transform offers great flexibility, in that, any number of functions satisfying some
basic constraints can serve as the parent wavelet in Equation 3.8. This flexibility has
permitted authors to tailor the shapes and time-frequency features of wavelets to match
the characteristics of signals they wish to detect, e.g., Wang (2001). The first condition
on a candidate parent wavelet g(t) is that the function g ∈ L2 (ℜ) satisfy an admissibility
condition
C g < +∞ .
(3.15)
This admissibility condition is necessary to guarantee that the inverse transform in
Equation 3.10 is valid. To guarantee this, the parent wavelet should be a zero mean
function such that
∞
∫ g (t )dt = 0
(3.16)
−∞
and thereby Gˆ (0) = 0 . By requiring that Gˆ (0) = 0 and that Gˆ ( f ) be continuously
differentiable, the admissibility condition is automatically satisfied. The condition of
continuous differentiability can readily be established for signals with sufficient time
decay
∞
∫ (1 + t ) g (t ) dt < +∞ .
(3.17)
−∞
In essence, this chain of properties helps to guarantee that the transform has finite support
in both the time and frequency domains, forming the “little waves” that gave rise to the
84
transform’s name. Note that, in some cases, the parent wavelet may also be normalized so
that g (t ) = 1 .
Parent wavelets satisfying these conditions may be subdivided into two classes:
real and analytic (or progressive), as discussed by Mallat (1998); however, only the
former class are presented herein, as they are the focus of the wavelet analyses in this
work. In general, an analytic signal is defined as a function that does not have energy
content at negative frequencies, i.e. has Fourier coefficients of zero at these frequencies.
Though complex in nature, analytic wavelets can be entirely characterized by their real
component. By definition, the lower limits of integration with respect to frequency and
scale in Equations 3.13 and 3.14 can be set to zero for this class of parent wavelets. The
suppression of negative frequencies also introduces a factor of two to these integral
relations, as shown in Mallat (1998).
3.5 Wavelet Resolutions: The Heisenberg Box
The scalogram in Equation 3.12 represents the level of energy within a box in the timefrequency domain, called a Heisenberg box. This box is centered at time t and frequency f
and has dimensions that are scaled versions of the time (∆tg) and frequency (∆fg)
resolutions of the parent wavelet
∆t = a∆t g
and
The box dimensions are then given by
85
∆f = ∆f g / a .
(3.18 a, b)
[t − a∆t , t + a∆t ]× ⎡⎢ f − ∆f
g
g
⎣
g
a
,f +
∆f g
⎤
a ⎥⎦
(3.19)
giving the box a span of 2∆t in the time domain and 2∆f in the frequency domain. Note
that the area of this box remains constant, 4∆t∆f, though the resolutions are a function of
scale. Thus, the Wavelet Transform gives local information about the function x(t) over
the
time
interval [t − a∆t g , t + a∆t g ]
∆f g
∆f g
⎡
f
−
f
+
,
⎢⎣
a
a
and
over
the
frequency
interval
⎤
⎥⎦ .
These basic relations demonstrate that the Heisenberg box of the wavelet shrinks
in the frequency domain at low frequencies (large scales) and shrinks in the time domain
at high frequencies (small scales). As shown in Figure 3.6, this makes the transform
better suited to capture the details of signal content at those frequencies. Though a host of
time-frequency transformations are available, only the Wavelet Transform is uniquely
capable of adaptively adjusting to the Heisenberg Uncertainty Principle. In essence, the
Wavelet Transform concedes that arbitrarily good resolution in both time and frequency
is not possible. Thus the transform optimizes its resolutions as needed. It provides good
resolution at high dilations or low frequencies, ideal for Civil Engineering applications,
while sacrificing its time resolution in this regime to satisfy the Uncertainty Principle. In
the time domain, the transform has good resolution at high frequencies in order to
identify signal singularities or discontinuities, a feature often exploited in health
monitoring applications, e.g. Corbin et al. (2000).
86
f
f2
2∆fg/a2
2∆tga2
2∆tga1
2∆fg/a1
f1
t
t2
t1
FIGURE 3.6. Concept of Heisenberg box for Wavelet
Transforms
Physically, the measure provided in Equation 3.18a,b indicates that two pulses in
time cannot be identified unless they are more than ∆t apart. Similarly, two distinct
frequency components cannot be discerned unless they are more than ∆f apart.
3.6 Discretization and Orthogonality
Frame theory establishes the completeness, stability and redundancy of linear discrete
signal representations, providing conditions for discretizing windowed Fourier
Transforms and Wavelet Transforms to retain a complete and stable, though potentially
redundant, representation (Carmona et al., 1998; Mallat, 1998). A complete
representation can be obtained only if the Heisenberg boxes of the transform fully cover
87
the time-frequency plane. Thus the sampling of the time-frequency space is based on the
time-frequency spread of the window w(t) in windowed Fourier Transforms or of the
parent wavelet g(t) in Wavelet Transforms. As long as the entire time-frequency plane is
covered by the Heisenberg boxes tied to each time and frequency/scale, the
representation is complete, though it still may be redundant and unstable in
reconstruction.
In moving toward a stable representation, the scales are often sampled along an
exponential sequence due to inverse relationship with frequency, according to the
bandwidth of the parent wavelet, while the time translations are sampled uniformly at
intervals proportional to the discretized scales. However, conditions introduced in Mallat
(1998) provide for a complete and stable translation-invariant representation, a capability
crucial in pattern recognition by discretizing only the frequency (scale). These common
dyadic Wavelet Transforms sample at v intermediate scales, or voices, over each octave
[2 ,2 ] and leave the time variable untouched. This leads to a dyadic scale sequence
j +1
j
{a }
j
j <0
1
for a = 2 v . The dyadic discrete Wavelet Transform (DWT) is then defined for a
signal x[n] of length N as
W [a j , n ] =
1
aj
N −1
∑ x[m]g
m =0
∗
⎡m − n⎤
⎢⎣ a j ⎥⎦ .
(3.20)
This expression of the discrete Wavelet Transform enables calculation by fast filter bank
algorithms. More importantly, when satisfying specific conditions of tight frames, entire
classes of orthogonal wavelets can be developed, as discussed in Carmona et al. (1998)
88
and Mallat (1998). Common orthogonal wavelets include Haar, Meyer and Daubechies
class. While beyond the scope of this work, these wavelets are tailored for image/signal
compression and reconstruction, though their time translation variance limits their usage
for pattern recognition (Wang, 2001), in which case the same transient information at a
different time may manifest different patterns in the time-frequency domain (Zheng et al.,
2002), motivating the use of the above mentioned dyadic transform. The use of the DWT
is furthermore not feasible for many of these investigations due to the limitations in
frequency and time resolution due to the sparse sampling of time-frequency space to
create tight frames. For this reason, Continuous Wavelet Transforms are advocated in this
work and the proper conditions for their discretization are provided in Chapter 4.
3.7 Wavelet-Transforms and Analytic Signals
Although the notions of TFDs have been utilized in signal processing for decades to
examine the variations in signal frequency with time, their extension to the identification
of the dynamic properties of mechanical and civil structures has evolved only recently.
The foundations for these analyses are deeply rooted in analytic signal theory and the
properties of phase and amplitude in complex signals, which are briefly overviewed
herein.
3.7.1 Instantaneous Frequency
A real signal x(t) may be represented as an amplitude A(t) modulated with a time-varying
phase φ(t)
89
x(t ) = A(t ) cos φ (t ) .
(3.21)
The instantaneous frequency (IF) is defined as the positive derivative of the phase
IF (t ) =
1 d
φ (t ) .
2π dt
(3.22)
While the notion of frequency at an instant may seem paradoxical, the simplest way to
conceptualize the notion of an “instantaneous” frequency is as the frequency of a sine
wave that locally fits the signal under consideration.
There are indefinitely many amplitude/phase pairs that can represent the signal in
the form of Equation 3.21. For example, the signal could be represented as a frequencymodulated signal with constant amplitude or an amplitude-modulated signal with
constant frequency. Amongst these pairs, there is one amplitude and phase pair that has
many desirable properties. This pair is determined by defining an analytic signal, z(t),
through a linear filtering operation that suppresses all negative frequencies. It was
proposed by Gabor (1946) to generate this complex analytic signal using a filter that
simply shifts the phases all frequency components by –π/2
z (t ) = x (t ) + iH [ x (t )] .
(3.23)
The operator H [⋅] denotes the Hilbert Transform, defined mathematically as
∞
H [ s (t )] =
∫
−∞
x (t − τ )
πτ
.
(3.24)
The operation denoted in Equation 3.24 is merely the convolution of x(t) with 1/t, or it
can be thought of as a phase shift of the signal by 90o. As a result, x(t) and H[x(t)] are
said to be in quadrature. This complex signal can then be completely characterized by
90
z (t ) = Az (t )e iφ z ( t )
(3.25)
with nonnegative amplitude Az and phase φz with values in the interval [0,2π), forming
the so-called canonical pair.
Since Re[z (t )] = x(t )
x(t ) = Az (t ) cos φ z (t )
(3.26)
yielding the canonical representation of x(t). Based on this representation, Ville (1948)
proposed to calculate the instantaneous frequency of the signal x(t) using the analytic
signal z(t)
IF (t ) =
1 d
1 d
[∠z (t )] .
φ z (t ) =
2π dt
2π dt
(3.27)
Note that the work in this field by Ville (1948) then led to his definition of an
instantaneous spectrum describing the spread of frequencies about this mean frequency,
known as the Wigner-Ville Distribution (WVD). However, for real signals, WVD
produces “ghost terms” as a consequence of the interference between positive and
negative frequencies (Carmona et al., 1998). It has also been observed that the WVD can
even exhibit negative bandwidth spectra (Jones & Boashash, 1990). Such lack of physical
significance to measures derived from the WVD led to some concerns for its practical
application. The measure in Equation 3.25 has physical significance only when φz(t) and
Az(t) meet some basic conditions associated with asymptotic signals, discussed in the
following section.
91
3.7.2 Theory of Asymptotic Signals
The class of asymptotic signals can be represented by the amplitude and phase of their
analytic counterparts. To identify if such a representation can be made, the following
condition is stated in Carmona et al. (1998)
−3
z (t ) = A(t )e iλφ ( t ) + O⎛⎜ λ 2 ⎞⎟
⎠
⎝
as λ → ∞
(3.28)
where λ is a positive number. This statement infers that the analytic signal approximation
becomes increasingly valid as the phase variations increase relative to the amplitude
variations. This characteristic distinguishes asymptotic signals and permits the analytic
signal approximation
z (t ) ≈ A(t )e iλφ (t ) .
(3.29)
In practical applications, the term λ is not known; however, Equation 3.28 nevertheless
places a very loose condition that the oscillations coming from the phase term should be
much faster than those coming from the amplitude term. This criterion may be best
imagined by viewing the amplitude and phase components of the signal in the Fourier
domain. The Fourier spectra of the amplitude (SA(f)) and phase (Sφ(f)) must be well
separated to satisfy this condition, as shown schematically in Figure 3.7.
Satisfaction of the asymptotic signal assumption has special benefits for analytic
Wavelet Transforms. As proven in Carmona et al. (1998), for the class of asymptotic
signals, the analytic Wavelet Transform coefficients at the stationary points are directly
proportional to the analytic signal z(t) plus an error term again on the order of λ-3/2. The
92
SA(f)
Sφ(f)
Sφ(f)
phase of
nonasymptotic
signal
phase of
asymptotic
signal
Frequency [Hz]
FIGURE 3.7. Schematic representation of asymptotic
signal conditions
stationary points are time-scale ordinates at which the frequency of the scaled wavelet
coincides with the local frequency of the signal. These findings indicate that as a signal
more strictly meets the asymptotic signal assumption, as λ → ∞, i.e. as the oscillations of
the phase term increase relative to the amplitude term, the better the wavelet coefficients
in the vicinity of the stationary points approximate the analytic signal. This finding then
allows analytic wavelet coefficients at these stationary points to be used directly to
estimate the analytic signal in Equation 3.25 instead of the Hilbert Transform as in
Equation 3.23. The merits of this development will be highlighted in a later section.
3.7.3 Theory of Wavelet Ridges
In cases for which these asymptotic signal assumptions are met, the Wavelet Transform
coefficients have some attractive characteristics in the vicinity of the aforementioned
stationary points. At these locations, the frequency of the scaled wavelet coincides with
the local frequency of the signal at that time. This similitude yields very large wavelet
93
coefficients at the scales associated with these frequencies, reflecting the property of
analytic wavelets to “concentrate” at the dominant frequencies within an asymptotic
signal. This pronounced localization is the hallmark of asymptotic signals in the timefrequency domain. The scales at which these concentrations occur forms a ridge in the
time-frequency plane described mathematically by
a r (t ) =
ωo
2πf o
=
φ ′(t ) IF (t )
(3.30)
where φ ′(t ) is the derivative of the phase and ωo (or fo) is the central frequency of the
parent wavelet, i.e. the frequency at which the Fourier Transform of the parent wavelet
focuses. This relationship indicates that once the scales associated with ridges are
identified, they can be used to define the analytic signal directly since
W ( a r (t ), t ) ∝ z (t )
(3.31)
and extract the instantaneous frequency of the signal, provided that there are no
significant noise components corrupting the signal (Carmona et al., 1998). Wavelet
coefficients along these ridges form the wavelet skeleton. The precise methodologies for
the extraction of wavelet ridges will be overviewed in Chapter 4.
3.7.4 Asymptotic Signals and Mechanical Oscillators
In the simplest structural identification problem of free vibration or impulse response, the
system has a distinguishable enveloped sinusoidal behavior given by
h(t ) = Ao e −ξω nt cos(ω n 1 − ξ 2 t ) = Ao e −ξω nt cos(ω D t )
94
(3.32)
where Ao is the initial displacement condition and θ is a phase shift. The analytic signal is
then
z (t ) = Ao e −ξω nt +iω Dt .
The amplitude function is given by
A(t ) = z (t ) = Ao e −ξωnt
with phase described by
φ (t ) = ∠z (t ) = ω D t .
(3.33)
(3.34)
(3.35)
Thus the damped natural frequency can be determined from the derivative of the phase,
equivalently viewed as the instantaneous frequency in Equation 3.27. Assuming a lightly
damped system, where ω D ≈ ω n , the damping can be determined from Equation 3.34.
The above relations are of course contingent upon the asymptotic signal assumption
being met, with the phase variations occurring more rapidly than those of the amplitude.
Later in Chapter 7, a specific example demonstrates the implications of violating this
condition for mechanical oscillators. If the frequency and damping terms in the above
relations are time-varying, the identification of instantaneous frequency and damping
values for a nonlinear system can similarly be accomplished using Equations 3.34 and
3.35. Feldman (1994a,b) advocated this more traditional approach using Hilbert
Transforms to generate the analytic signal in Equation 3.33. However, this transform has
inherent limitations, explored in the following section.
3.7.5 Alternative Methods to Generate Analytic Signals for Multi-Component Data
Despite its wide usage, the Hilbert Transform is especially sensitive to the asymptotic
signal restrictions. The extent to which this condition is satisfied dictates whether the
95
transform truly yields the quadrature component. Breakdowns were demonstrated in the
free vibration of mechanical oscillators with low natural frequencies as damping was
increased (Ruzzene et al., 1997). To circumvent this, other transforms were proposed that
would directly yield a complex representation without the approximation in Equation
3.23. Transforms such as the Wigner-Ville Distribution (Feldman & Braun, 1995) were
employed; however they were limited by the restriction of monocomponent signals, much
like the Hilbert Transform.
There is considerable debate surrounding how to quantify the conditions under
which the Hilbert Transform can be used to obtain the analytic signal in Equation 3.25
and allow Equation 3.27 to yield a physically meaningful instantaneous frequency. The
leakage of signal energy into negative frequencies of the Fourier spectrum constitutes one
condition under which the Hilbert Transform will not yield the quadrature component
(Boashash, 1992a), as analytic signals suppress negative frequencies; however, a more
concerning issue surrounds its inability to handle signals with multiple frequency
components. To demonstrate, consider a signal with several distinct harmonic
components, e.g. a summation of several sine waves of varying frequency. Despite the
presence of several distinct frequencies, the direct application of the Hilbert Transform to
extract the instantaneous frequency in the aforementioned strategy will yield a single
estimate of instantaneous frequency at each instant of time that is an average or
composite of all the individual frequency components present. This inability to directly
distinguish multiple components leads to this "monocomponent" restriction, necessitating
that there be only a single frequency component present in the signal at each time.
96
Therefore, prior to the application of the Hilbert Transform, "multi-component" signals
must undergo some type of bandpass filtering operation.
Understandably, this “monocomponent” or “multi-component” terminology arises
from a Fourier perspective, appealing to those familiar with modal structural analysis
backgrounds. Unfortunately, there is no clear criteria to determine if a given signal is
truly multi-component or monocomponent, despite efforts by Boashash (1992a) defining
multi-component signals as those that may be comprised by a weighted sum of
monocomponent signals. Cohen & Lee (1989) used this same premise to propose similar
methods to identify multi-component signals. However, the lack of a precise means to
distinguish multi-component signals led to the adoption of a "narrowband" condition
(Schwartz et al., 1966). Even with this definition, the manner in which narrowbandedness
is determined is subjective, leaving the possibility that signals meeting a specified
narrowband condition may still contain multiple frequency components at a given instant
in time, as the example in Section 7.5.5 will later demonstrate. Thus, while the debate
over correctness of terminology still ensues, the signal processing community in general
concurs that the more closely a signal approaches a strictly narrowband condition, the
better the Hilbert Transform approximates the quadrature signal and thereby the
instantaneous frequency (Boashash, 1992a). For the purposes of this work, the
terminology of monocomponent and multi-component signals will be retained to
respectively describe signals that have one or multiple distinct frequencies at each instant
in time.
97
The bandpassing process required for multi-component signals can be difficult for
closely spaced modes and must be addressed when using the Hilbert Transform (Lee &
Park, 1994) or the Wigner-Ville Distribution (Feldman & Braun, 1995). A viable multicomponent decoupling approach, Empirical Mode Decomposition, used in conjunction
with the Hilbert Transform, was proposed by Huang et al. (1998) and will be discussed in
further detail in Chapter 7. The use of this approach has permitted system identification
of large structures using the Hilbert Transform analytic signal approach (Yang et al.,
2000). However, the Wavelet Transform provides an alternative analysis framework that
directly produces a complex analytical signal and separates multiple components,
circumventing the need for bandpass filtering. Recognizing the ability of wavelet
transforms to decouple multi-component signals, a number of researchers began applying
this technique in various forms for analysis of impulse (Huang et al,. 1994; Robertson et
al., 1998a,b) and free vibration response (Hans et al., 2000; Lamarque et al., 2000;
Ruzzene et al., 1997; Staszewski, 1997). The three latter studies drew upon the unique
characteristics of Morlet wavelets and the relationships adapted Equations 3.34 and 3.35
for wavelet system identification in linear and nonlinear systems, with wavelet amplitude
and phase being proportional to their counterparts in the analytic signal along wavelet
ridges.
3.8 Benefits of the Wavelet Transform for Civil Engineering Applications
This chapter chronicled the evolution of time-frequency analysis and overviewed the
mathematical theory governing Wavelet Transforms and their relationship to asymptotic
signals. The wavelet’s ability to uncover local and transient features, while implicitly
98
decoupling multiple frequency components, makes it a system identification and analysis
tool well suited to a variety of problems in Civil Engineering, including the study of
earthquake, wind and wave-induced response (Gurley & Kareem, 1999). For example, in
the case of earthquakes, the onset of damage can be identified in time by the lengthening
of the structural period, or shortening of natural frequency in the wavelet map, and
related to the characteristics of the ground motion at that instant. Clearly such insights
could not be gained through traditional Fourier techniques, as subsequent chapters will
demonstrate in the analysis of a variety of simulated and measured signals. However,
before such analyses can be conducted, a number of processing issues must be addressed
to insure the proper application of the transform and meaningful interpretation of results.
The wavelet analysis framework in Chapter 4 addresses these issues.
99
CHAPTER 4
FRAMEWORK FOR WAVELET-BASED ANALYSIS
4.1 Necessity for Unified Analysis Framework
The use of wavelet-based analysis in a variety of fields previously focused largely on
qualitative measures of time-varying frequency content, during which signal processing
concerns were not previously raised. However, the adaptation of the transform to more
quantitative analyses, including system identification, necessitates that these processing
concerns be explored and remedied. Just as years of Fourier Transform applications have
uncovered issues associated with leakage, aliasing and frequency resolution, investigation
into such phenomenon for wavelet-based analysis is lacking. One component of this
research effort was to develop a complete framework for wavelet analysis to address
these concerns and insure the most accurate wavelet coefficients possible. This chapter
first presents the parent wavelet that is used throughout this research and then introduces
each of the processing considerations associated with this wavelet and, based upon the
properties of this parent wavelet, provides some appropriate remedies. This framework
provides the backdrop for this research, creating a single framework in which this class of
wavelet can be extended to Civil Engineering applications.
100
4.2 Morlet Wavelets
The great flexibility in the transform leads to a variety of parent wavelets, each providing
a different scalogram representation for a given signal. This dependence on the choice of
parent wavelet has been criticized by some, e.g., Huang et al. (1998). However, others
counter that the parent wavelet should be selected based upon shape and form of the
signal in question or a particular feature to be extracted, making this flexibility an asset.
Wang (2001) demonstrates the significance of selecting parent wavelets that resemble
assorted transients in mechanical systems and using this ability to zoom in on a specific
signal feature. While there is a wide array of mother wavelets available for analysis, a
particular class of wavelet is best suited for the purposes of this study. The Morlet
wavelet, an approximately analytic wavelet, is merely the frequency modulation of a real
and symmetric window function — the Gaussian window. The Morlet Wavelet
(Grossman and Morlet, 1984) can be viewed as a Gaussian-windowed Fourier Transform
g (t ) = e
−t
2
/2
e j 2πf o t = e
−t
2
/2
(cos(2πf ot ) + j sin( 2πf ot )) .
(4.1)
whose sine and cosine basis functions oscillate at the central frequency fo. For this reason,
this wavelet is sometimes called the Gabor wavelet. Dilations of this temporally localized
parent wavelet then allow the “effective frequency” of this sine-cosine pair to change in
order to match harmonic components within the signal. Its basic analogs to the Fourier
Transform, shown in Figure 4.1, make it well suited to for harmonic analysis in the timefrequency domain. This wavelet approximately satisfies the conditions placed on parent
wavelets in Section 3.4.2 for sufficiently large values of fo., e.g. greater than 5/(2π) or 0.8
(Carmona et al., 1998).
101
Wavelet Basis: Morlet Wavelet
Fourier Basis: Cosine Function
Re [g(t)]
Re [g(t)]
g ( t ) = e iω o t e − t
= e −t
2
/2
2
g (t ) = e iω o t
= cos(ω o t ) + i sin(ω o t )
/2
(cos(ω o t ) + i sin(ω o t ) )
FIGURE 4.1. Comparison of basis functions for Fourier and
Morlet Wavelet Transforms
The Morlet wavelet is equivalently localized in the frequency domain, as
evidenced by the Fourier Transform of the dilated Morlet wavelet
G (af ) = 2 π e −2π
2
( af − f o ) 2
.
(4.2)
In the case of the Morlet Wavelet, there is a unique relationship between the dilation
parameter of the transform, the scale a, and the Fourier frequency f at which the wavelet
is focused. This relationship is evident by maximizing Equation 4.2 to yield:
a=
fo
f
.
(4.3)
The direct relationship to Fourier frequency and analogs to the Fourier Transform itself
makes the Morlet wavelet analysis quite natural for interpretation by engineers already
versed in Fourier analysis. Note that the relation between Fourier frequency and scale can
be approximately estimated for some other parent wavelets, as discussed in Meyers et al.
(1993), though few can deliver as direct a relation as that shown in Equation 4.3.
102
4.2.1 Frequency Resolutions: Tuning the Central Frequency
While the relationship for wavelet frequency resolution in Equation 3.18b may seem
straightforward, for the Morlet wavelet the issue of resolution is complicated by the fact
that it is a Gaussian-windowed Fourier Transform. Unfortunately, this window lacks
measurable duration. An “effective duration” can be defined by drawing on the common
use of the Gaussian window in both the Short-Time Fourier Transform and the Morlet
wavelet. Thus the mean-square definition for frequency duration given by Equation 3.6
can similarly be used to yield (Chui, 1992)
∆f g =
1
2π 2
.
(4.4)
Combining this with Equations 3.18b and 4.3, the resolution of the Morlet wavelet at a
given frequency fi is given by
∆f i =
fi
.
2π 2 f o
(4.5)
This not only dictates that the frequency resolution will improve for longer period
signals, but also implies that the resolution capabilities can be appropriately adjusted for
the analysis through careful selection of the central frequency of the Morlet wavelet, a
fact emphasized throughout this research and by Wooh & Veroy (2001). As pointed out
by Cohen (1999), this is a delicate matter that can lead to significant distortion of the
signal if the trade offs between time and frequency resolution are not properly balanced.
Intuitively, as this central frequency increases, the frequency resolution is enhanced as
more cycles are condensed within the localized window, as shown in Figure 4.2. The
analyses throughout this study exploit this flexibility to “tune” the wavelet analysis to
103
FIGURE 4.2. Real and imaginary components of Morlet wavelet for
various central frequencies
extract the desired characteristics of the signal under consideration, conducting analyses
in tiers to progressively refine the frequency resolution. In Chapters 5 and 7, examples
demonstrate the notions of primary and secondary wavelet analyses with increasing
central frequency. The central frequency can also be varied over an entire suite of values
by treating fo as a third variable in the Wavelet Transform, in essence creating a fourdimensional view of wavelet coefficients varying with time, scale and central frequency,
as proposed in Wang (2001); however, such a rigorous analysis is not required if the
required value of the parameter can be determined a priori using a simple inspection of
the time series.
104
In particular, this research has explored the issue of closely spaced frequency
components and the importance of central frequency tuning, as shown later in Section
7.5.5, making the following observations. To achieve complete separation, it should be
first noted that the bandwidth (or width of the Heisenberg box) associated with a given
frequency in a Wavelet Transform is 2∆fi. However, by basing the bandwidth measure on
Gabor’s mean square estimate for the Gaussian window (see Equation 4.4), only about
68% of the frequency window is accounted for. This measure essentially defines one
standard deviation (σ) of the window mean, yielding a frequency window with a total
bandwidth 2∆f or 2σ, as shown in Figure 4.3. However, the duration of this frequency
window is misleadingly narrow, assuming a better frequency resolution than is truly
present. To more strictly define the bandwidth, one should note that roughly 95% of the
window lies within 2 standard deviations of the mean, as also noted in Figure 4.3, for
which the effective bandwidth is 4∆f or 4σ. An even stricter condition would extend the
definition to 3 standard deviations, also shown in Figure 4.3, encompassing 99% of the
window. These considerations become important when confronted with the need to
separate closely spaced modes, as demonstrated in the analyses in Chapter 8.
Therefore, Equation 4.5 may be generalized, given the need to separate two
closely spaced frequency components f1 and f2, where the minimum central frequency for
an analysis can be determined by
f o = (2α )
f1, 2
2π 2∆f1, 2
105
(4.6)
f
2∆fi
fi+3∆fi
fi+2∆fi
fi+∆fi
fi
fi-∆fi
fi-2∆fi
fi-3∆fi
tj-3∆ti
tj-2∆ti
tj-∆ti
tj
tj+∆ti
tj+2∆ti
tj+3∆ti
2∆ti
t
FIGURE 4.3. Effective frequency and time resolutions
of Morlet wavelet and associated Heisenberg box
f1,2 can be taken as either f1 or f2 or an average of the two, as the choice has little impact
for small values of ∆f1,2, the separation between the two frequency components. α is the
parameter defining how much overlap between the Gaussian analysis windows at
adjacent frequencies is permitted. As shown in Figure 4.4, using the traditional mean
square definition for the bandwidth of the Gaussian window amounts to α = 1. As
determined by Equation 4.6, the wavelet analysis windows centered at the two
frequencies will overlap significantly. However, increasing α to 2 and applying Equation
4.6, a larger central frequency is adopted, yielding analysis windows that are narrower,
and though centered at the same two frequencies, now has reduced overlap. Note that α
can be increased to 3, insuring complete modal separation. As will be shown later in
106
∆f1,2
α=1
∆f1
∆f2
∆f1
∆f2
f1
f2
α=2
FIGURE 4.4. Schematic demonstration the implication of
parameter selection for modal separation
Chapter 8, retaining α = 2 is typically sufficient to insure adequate modal separation for
system identification in most linear systems, though total modal separation via α = 3 may
be necessary for closely spaced modes when nonlinear system identification is
performed.
4.2.2 Temporal Resolution: A Physical Understanding of End Effects
Similarly, the temporal resolution of the Morlet wavelet is dictated by the Gaussian
window, defined using the aforementioned mean square definition (Chui, 1992)
107
∆t g =
1
.
2
(4.7)
As mentioned in the discussion of the STFT in Chapter 3, the uncertainty principle
dictates that the minimum value of the time and frequency resolution product is 1/4π,
giving the Morlet wavelet the best possible time and frequency resolutions.
The temporal resolutions of a wavelet analysis have direct bearing on the
significance of end effects in Wavelet-Transformed data, which have been noted in a
number of applications, e.g. Staszewski (1998). The presence of end effects in wavelet
analyses was previously credited to many familiar numerical by-products of Fourier
Transforms, as they are used to calculate Wavelet Transforms according to Equation 3.9.
As the Fourier Transform assumes periodicity of the time series, Torrence & Compo
(1998) proposed the familiar tactic of zero padding the beginning and end of the time
series, lengthening the time series to the nearest power of two to speed calculations by
FFT. However, the discontinuity potentially created at the signal ends by the sudden
transition to zero values in padded regions manifests itself as a smearing of energy in the
wavelet time-frequency plane. While Torrence & Compo (1998) quantify the cone of
influence to define the region where end effects are prevalent, they proposed no formal
solution. An alternative remedy is to create a smoother transition to zero end values, time
series tapering using windowing functions has also been proposed (Meyers et al., 1993),
though the modification of signal energy as a result of the windowing function is
unattractive. However, the Gaussian window on the Fourier basis of the Morlet Wavelet
Transform already eliminates the more classical problem that discontinuities and a lack of
108
periodicity present for Fourier-based numerical implementations, implying that the
persistent end effects in Morlet Wavelet-Transformed data have another source.
Though the cone of influence allows the quantification of end effects regions, it
offers no remedy, leading to the corruption of considerable amounts of wavelet
coefficients. This issue is addressed in this work through the following developments and
the introduction of a reflective padding scheme. To revisit this issue, examine the
convolution operation in Equation 3.8 and the parent wavelet in Equation 4.1. It is
evident that, although the wavelet is focused at a given time and represents the temporal
content in this vicinity, the window extends into both the past and future (see Figure 4.3),
by an extent dependent on the scale being analyzed. The scaled time resolution of the
wavelet is given by
∆t i =
fo
fi 2
.
(4.8)
As illustrated by Figure 4.5, the temporal window is elongated at the lower
frequency, f1. As the analysis time tj lies within ∆t1 of the ends of the signal, the analysis
window extends into a region with no available data, yielding wavelet coefficients based
on an “incomplete” signal. Intuitively, it is reasonable to then question the accuracy of
wavelet-transformed data within ∆ti of the beginning and end of the signal.
Once again, this resolution in Equation 4.8 is based on the classical mean square
definition in Equation 3.5, approximating the effective temporal duration as a single
standard deviation of the Gaussian window. Examining Figure 4.6, in light of the
previous discussion with respect to frequency resolution, it is evident that the expression
109
f
f2
∆t2
∆t1
f1
t
tj
FIGURE 4.5. Illustration
susceptible to end effects
of
region
g(t)
∆ti
2 ∆ti
3∆ti
4∆ti
t [s]
FIGURE 4.6. Real component of Morlet wavelet with Gaussian window
emphasized
110
in Equation 4.8 accounts only for 68% of the total window area. A more stringent
condition may be obtained by defining the effective temporal resolution of the Morlet
wavelet as 2σ or 2∆ti, in order to account for about 95% of the Gaussian window.
Defining a 3∆ti as the temporal resolution can impose an even more stringent condition.
Dependent on the desired level of accuracy, either of these conditions can be
imposed to quantify the regions potentially at risk to end effects. The general expression
β∆ti ≤ t j ≤ T − β∆ti ,
(4.9)
where β can be set to any positive, nonzero value, provides a practical limit on the
translations t that can be considered in a wavelet analysis of a signal of length T. Thus,
only wavelet coefficients satisfying Equation 4.9 can be reliably analyzed. Equation 4.8
dictates that meaningful analyses at low frequencies require larger amounts of data, so
that several cycles of the low frequency phenomena can be retained.
4.2.2.1
End Effects: Influence of End Effects on Spectral Amplitude
For a simple illustration of the implications of end effects on spectral amplitude, consider
a sine wave with frequency fn (taken as 1 Hz). In theory, the Morlet Wavelet Transform
of this signal yields a scalogram that is constant with time
W (a, t ) = 2π 3 ae − 4π
2
2
( af n − f o ) 2
.
(4.10)
The values of the wavelet scalogram, at a given time, can be plotted to yield an
instantaneous power spectrum. Equation 4.10 equivalently provides this for the simple
sinusoidal signal.
111
Performing the Morlet Wavelet Transform (fo = 2 Hz) on the sinusoid yields a
time-varying scalogram, in contrast to the anticipated result. The evolution of the
scalogram with time can be examined by plotting the scalogram values along the wavelet
ridge. This result, shown in Figure 4.7, displays a rounding of what should be a constant
scalogram coefficient. The degree of deviation from the theoretical result, shown as the
dotted line, becomes less marked in the interior of the signal. The vertical bars denote the
end effects regions defined in Equation 4.9 for various values of β and indicate that the
calculated Wavelet Transform will more accurately approach the theoretical result for
5∆ti
4∆ti
3∆ti
2∆ti
|W(a,t)|2
2∆ti
3∆ti
4∆ti
5∆ti
β > 2.
t [s]
FIGURE 4.7. Scalogram of sine wave along ridge. Dotted line denotes theory
and solid line is calculated result. Vertical bars demarcate end effects regions,
β∆ti, for β=1-5
Figure 4.8 further illustrates the ramifications of analyzing instantaneous power
spectra taken from end effects regions. The calculated instantaneous spectra at each time
are plotted one atop the other in gray, essentially collapsing the scalogram in time.
According to Equation 4.10, there should be no variation among them. However, by
112
|W(a,t)|2
β=0
f [Hz]
β=1
|W(a,t)|2
|W(a,t)|2
β=2
f [Hz]
f [Hz]
β=3
|W(a,t)|2
|W(a,t)|2
β=4
f [Hz]
f [Hz]
FIGURE 4.8. Deviations of simulated instantaneous spectra (gray) from
theoretical result (black) as end-effects regions are progressively neglected
113
including the spectra from end effects regions (β = 0), there is considerable variance in
the plot, as the spectra show notable deviations from Equation 4.10 shown in black. Note
that the deviations are more marked on the high frequency side of the spectrum, a result
of the lessened frequency resolution at lower scales. Through a more stringent condition,
increasing β in Equation 4.9, the neglected regions lengthen, the variance among the
spectra is reduced, and the deviations from the theoretical expression are minimized.
Note that even the commonly used definition of wavelet temporal duration (β = 1) is an
insufficient measure of the end effects region producing these deviant spectra.
Unfortunately, the use of larger β values reduces the amount of useable transformed data.
Thus, while β = 4 produces a sufficiently accurate means to quantify end effects regions
and separate deviant spectra, β = 3 is shown in Chapter 8 to be sufficient for most
analyses in terms of capturing accurately the spectral amplitude.
4.2.2.2
End Effects: Influence on Spectral Bandwidth Estimation
As a consequence of the windowing applied by the Gaussian function in the Morlet
wavelet, the bandwidth of the resulting wavelet instantaneous spectra are larger than their
Fourier equivalent. This gives the appearance of a larger value of effective damping in
the signal, the extent of which depends on the scale analyzed. Consider the Morlet
wavelet expression in the Fourier domain, given by Equation 4.2. The half-power
bandwidth introduced in Chapter 2 can be used to provide a simple measure of the
bandwidth of wavelet spectra. The HPBW is defined in this case as the two frequencies
that coincide to half the amplitude of the wavelet’s instantaneous spectral peak, shown as
f1 and f2 in Figure 4.9.
114
|W(a,t)|2
max[ |W(a,t)|2 ]
½ max[ |W(a,t)|2 ]
WT
B
f1
fi
f
f2
FIGURE 4.9. Schematic representation of
half-power bandwidth estimation
The HPBW is defined the difference between these two frequencies
BWT = f 2 − f1 .
(4.11)
The two frequencies corresponding to the half-power level of the Morlet spectra are
given by
ln(2)
2π
f = fo ±
(4.12)
yielding a half-power bandwidth of
Bmorl =
ln(2)
π
.
(4.13)
This is the half-power bandwidth of the Morlet wavelet. For a simple sine wave, the
Morlet Wavelet Transform will localize at the frequency of the sine wave, though the
bandwidth of the wavelet spectrum will be completely dictated by the parent wavelet, as
115
a sine in theory has no bandwidth.
Recalling that the resolutions of the Wavelet
Transform are merely scaled versions of the parent wavelet, this half-power bandwidth
measure in Equation 4.13 can be similarly scaled by dividing by a. Note that in the
definition of the half-power bandwidth, the spectra is assumed symmetrical. Due to the
multi-resolution nature of wavelets, wavelet spectra broaden toward the higher
frequencies, but for a narrowbanded spectrum, the assumption of symmetry can be
retained. Therefore, the scale at which this half-power bandwidth is evaluated should be
the scale defining the ridge of the transform, at which the signal energy is focused.
Understandably, this scale corresponds to the instantaneous frequency of the system, or in
this case, the frequency of a simple sine wave, fn. Thus the half-power bandwidth of a
wavelet-transformed sine wave is given by
BWT =
ln(2) f n
.
π
fo
(4.14)
As discussed in the previous example, deviations in terms of amplitude of the
instantaneous spectra were sufficiently mitigated by neglecting those spectra which were
generated in the end effects regions, defined by assuming β = 4. As shown in Figure 4.7,
the characteristic rounded amplitude deviations of end effects regions seem to diminish
after 4∆t. However, the influence of end effects on more sensitive spectral measures such
as the half-power bandwidth is not completely remedied by neglecting this end region.
The implications of not neglecting spectra drawn from end effects regions is shown in the
first image in Figure 4.10, revisiting the same signal from Section 4.2.2.1 for a more
refined wavelet analysis with fo=5 Hz. The calculated half-power bandwidth deviates
significantly at the ends of the signal from the theoretical result shown by the dashed line.
116
β=0
BWT [Hz]
BWT [Hz]
β=4
t [s]
t [s]
β=6
BWT [Hz]
BWT [Hz]
β=5
t [s]
t [s]
FIGURE 4.10. Improvements made in half-power bandwidth estimates by
successively neglecting larger end effects regions. Theoretical prediction
(dashed) and calculated result
Using the criteria of β = 4 to neglect regions defined by Equation 4.9 improves the result,
though the deviations from theory are still quite evident in the tails. By selecting more
stringent conditions on β, the deviations from theory are minimized and the bandwidth
estimated in the simulation takes on a constant value. For β = 6, the deviation between
theory (BWT = 0.053) and simulation (BWT = 0.053) are essentially identical. Though the
deviations in Figure 4.10 are easily explained by the end effects phenomenon, simply
neglecting these regions in analysis yields to a considerable loss of data, especially in the
case of bandwidth estimation, where for β = 6, only one-third of the transformed signal is
deemed reliable.
117
4.2.2.3
End Effects: Melioration via Reflective Padding
The loss of considerable regions of a signal is the unfortunate consequence of end effects.
Particularly for more sensitive measures like bandwidth, the loss of useable transformed
signal can be quite significant: approximately 10 times the effective duration of the
lowest frequency component of interest. Meyers et al. (1993) noted the presence of these
effects and compiled a number of strategies for their minimization. However, in these
discussions, the authors largely attributed the problem to one associated with the use of
Fourier transforms in calculating the wavelet transform. As a result, the authors presented
strategies directly from Fourier processing annals. The first remedy, the introduction of a
cosine window to preprocess the time history, essentially led to a loss of meaningful data.
The authors then proposed detrending the data and removing the mean to enhance
performance of the Fourier transform and found the technique to have no benefit in
mitigating wavelet end effects. Though these are common methods for Fourier
processing, they did not provide great utility for wavelets since they did not address the
aforementioned root cause: the wavelet’s assimilation of past, present and future
information in each analysis window. However, their final remedy to pad the data with a
short tail gradually approaching zero did focus inadvertently on this root cause. Though
never motivated in their work as such, the padding of the beginning and end of the signal
with surrogate values to place the true signal of interest at the center of the transformed
vector has promise. In so doing, the “dummy” values at the tails were allowed to be
corrupted by the end effects phenomenon, preserving the true data in question. Meyers et
al. (1993) chose this technique to force periodicity upon nonperiodic data, again
reflecting the authors’ theory that Fourier transforms were ultimately to blame. Though
118
this was a positive first step, the rather arbitrary tails of empirically defined length did not
preserve signal characteristics at the ends, nor provided recommendations on the length
of tails required to mitigate end effects corruption. In a later study, Torrence & Compo
(1998) attempted a similar elongation of a signal at its ends using zero padding; however,
it should be reiterated that the wavelet considers both past and present information at
each time step, so the addition of zero values still provides wavelet coefficients at the
ends that are derived in part from a “zero-value” signal. Though the greatest contribution
to a wavelet coefficient at that point in time comes from the signal immediately
surrounding that point, data displaced further in time are also considered to an extent.
Therefore the signal cannot be padded with zero values or arbitrary non-zero values as
suggested by Meyers et al. (1993), as analysis of the true signal near the beginning and
end of the original signal will certainly dip into these padded regions. Thus, the regions
should locally preserve the frequency and bandwidth characteristics of the signal.
This local preservation can be achieved by merely reflecting the signal about its
beginning and end, appropriately based on the odd or even character of the signal. If
continuity of the signal is not preserved in the transition to the padded region, the wavelet
will rightfully detect a discontinuity. Thus improper padding can equally introduce
further errors to Wavelet Transforms. To illustrate, consider a simple sine and cosine pair
shown Figure 4.11. To preserve continuity at the ends, even functions must be directly
reflected and odd functions must be reflected negatively. This same notion can be
extended to more complicated signals. Figure 4.12 illustrates an arbitrary signal and the
shaded regions that will potentially be useless following the Wavelet Transform.
119
EVEN REFLECTION
ODD REFLECTION
t [s]
t [s]
FIGURE 4.11. Even and odd signals and their respective reflections
(dotted line)
PADDED SIGNAL
ORIGINAL SIGNAL
β∆t
β∆t
β∆t
β∆t
T
T
FIGURE 4.12. Concept of signal padding for arbitrary function
Depending on the level of β and fo selected, these regions can consume two-thirds
or more of the signal. In the padding operation, the signal is elongated by 2β∆t as the
signal is continuously reflected about the start and end of the signal. Now the two shaded
regions envelop the dummy reflections of the signal, while entire duration of the true
signal is conserved and can be analyzed with little contamination from end effects.
120
Mathematically, this operation may be conducted by considering a signal x
described by
x = [x1
x2 K x N ]
(4.15)
with sampled time vector
t = [t1 K t N ].
(4.16)
The temporal duration of the analyzing wavelet is then determined using Equation
4.8 based on the lowest frequency being considered in the analysis (f1), as this frequency
will yield the longest duration ∆t1 and thereby dictates the maximum end effects
anticipated. β is then selected based on the desired accuracy of the resulting spectra, and
the time ordinates closest to the termination of the end effects regions are then identified
by
t 0 = min[t > β∆t1 ] and t m = max[t < (t N − β∆t1 )] .
(4.17)
The modified signal xMOD is constructed by reflecting the signal x (for even functions) or
its negative (for odd functions) for the duration of β∆t1 about t1 and tN, according to
x MOD = [± xn
± xn−1 K x1 K x N
± x N −1 K ± x0 ]
(4.18)
where xn and x0 are the values of the sampled signal x at time t0 and tm. xMOD is then
Wavelet Transformed and the coefficients calculated from the padded regions are simply
neglected, retaining only the coefficients of the true signal for meaningful analysis. This
pre-processing technique will be referred to herein as reflective padding.
121
4.2.2.4
End Effects: Efficacy of Reflective Padding Scheme
As sine waves are represented by wavelets in a very simplistic form, they are now used to
illustrate the efficacy of the proposed padding scheme. Recall that the wavelet
instantaneous spectrum of a simple sine wave does not vary in time. As shown by Figure
4.7, the scalogram of a simple sine, when viewed purely as a function of time, is a
constant. Deviations from that constant were shown to be the hallmark of end effects in
the Wavelet Transform. Thus any signal composed of a series of M sine waves, will yield
a scalogram containing M constant ridges in the time-frequency domain. The values of
the scalogram along each ridge can be individually examined for deviations from the
theoretical result. Fortunately, though this summation of sines is capable of generating
complicated time series, they are simply analyzed in the wavelet domain by virtue of its
inherent bandpass filtering. Therefore, the efficacy of the reflective padding scheme on a
complicated time series can be demonstrated by examining the simple behavior of the
component sines in the wavelet domain.
The following signal is generated for this example
x(t ) = ∑ sin(ω i t )
(4.19)
i
where ωι = 2π [0.28 Hz, 0.5 Hz, 0.7 Hz, 1.0 Hz, 1.4 Hz, 1.65 Hz, 1.9 Hz, 2.25 Hz, 2.7
Hz, 3.25 Hz]. The signal was simulated for 10 minutes, sampled at 10 Hz. Snapshots of
the first 100 s of the signal are shown in Figure 4.13. As a consequence of the multiresolution character of wavelets, at higher frequencies, the frequency resolution is
reduced. Thus, the last 6 harmonics may overlap one another considerably if a low
122
x(t)
x(t)
t [s]
x(t)
x(t)
t [s]
t [s]
t [s]
FIGURE 4.13. Simulated time series for validation of
padding scheme
frequency Morlet wavelet is chosen. Therefore, to enhance the frequency resolution and
separate closer-spaced, high frequency content, a central frequency of 10 Hz is chosen for
the analysis.
Following the calculation of the Wavelet Transform, the ten ridges are extracted
from the resulting scalogram. Though omitted for brevity, these plots display a
characteristic rounding as previously observed in Figure 4.7. It is evident that the end
effects regions, even for such a large central frequency, decrease significantly with
increasing frequency, as indicated by Equation 4.8. This may be the reason that previous
studies did not encounter significant manifestations of end effects, as most wavelet
analyses have been concerned with higher frequency mechanical systems. However, in
123
the study of low frequency Civil Engineering structures, the presence of end effects
should be anticipated and remedied, making the proposed padding procedure of essence.
The overlaying of the instantaneous spectra in time to collapse the scalogram
along that axis demonstrates the deviations that occur in these end regions, as shown in
Figure 4.14. The calculated wavelet spectra are shown in gray, with the theoretical
prediction in black. First note that the spectra again have a variable bandwidth that
decreases with frequency, as discussed previously. This is a consequence of the enhanced
frequency resolution of the wavelet for large scales. For the low frequency components,
the deviations are not as evident, due to the scaling of the plot. However, the deviations
are more marked for the high frequency components due to the lack of frequency
resolution. Although the end effects regions for the higher frequency modes are not as
lengthy, the deviations of the few spectra taken from these regions are considerable,
especially in the case of the 10th component. It is evident that the 10 Hz Morlet wavelet
was capable of separating the modes of the system. Note that the quality of Figure 4.14
could have been enhanced by simply neglecting the spectra that were derived from end
effects regions, however that results in a loss of a significant amount of data.
The lowest harmonic component produced the largest end effects region in the
wavelet skeleton, approximately 100 seconds of corrupted coefficients. Thus, this value,
at minimum, must be used to pad the signal, as discussed previously. The success of the
padding operation is first gauged by looking at Figure 4.15, an overlay of the
instantaneous spectra, as was shown for the unpadded case in Figure 4.14. In this case,
the spectra generated from the Wavelet Transform of the modified signal in Equation
124
|W(a,t)|2
f [Hz]
FIGURE 4.14. Superposition of instantaneous spectra over all time for
calculated Wavelet Transform (gray) with theoretical prediction (black)
4.18 are plotted. Only spectra obtained from the true signal are plotted and all those
determined from the virtual values of the padded signal are discarded. Note that there is
no discernable difference between the predicted and calculated results when assuming β
= 4.
In terms of bandwidth estimates, the half-power bandwidth’s accuracy is
enhanced in these end regions when the padding scheme is employed. For brevity, a
demonstration is provided using only three of the modes. Figure 4.16 displays the
unpadded bandwidth measures, demonstrating the characteristic trademark of end effects.
Note again that the portions of the signal lost due to end effects is more marked at lower
125
|W(a,t)|2
f [Hz]
FIGURE 4.15. Superposition of instantaneous spectra over all time for
calculated Wavelet Transform (gray) with theoretical prediction (black)
with padding operation
frequencies, where nearly one-third of the values have been compromised. As
demonstrated in Figure 4.16, β = 6 is a necessary condition to remove all traces of end
effects in the bandwidth measure. Using this condition in conjunction with the padding
operation, a precise definition of the half-power bandwidth is maintained over the entire
duration of the signal, as shown in Figure 4.16d-f. The simulated bandwidth measure is
less than 0.2% of the theoretical prediction for all of the modes in this example.
Although padding the signal with itself insures that the spectral content of
surrogate regions locally matches that of the true signal, this should not be viewed as a
126
BWT [Hz]
BWT [Hz]
BWT [Hz]
(a)
(d)
(b)
(e)
(c)
(f)
Time (s)
Time (s)
FIGURE 4.16. Efficiacy of padding operation to reduce end effects in wavelet
bandwidth measures, theoretical prediction (dashed) and simulation (solid): (a)
first mode half-power bandwidth without padding, (b) fifth mode half-power
bandwidth without padding, (c) tenth mode half-power bandwidth without
padding, (d) first mode half-power bandwidth with padding and β = 6, (e) fifth
mode half-power bandwidth with padding and β = 6, (f) tenth mode half-power
bandwidth with padding and β = 6
127
way to defeat the Heisenberg Uncertainty Principle. It should be reiterated that the end
effects are merely a physical manifestation of the wavelet’s inherent analysis windows,
which lengthen as fo is increased. Although the end effects can be repaired, the larger
temporal analysis windows imply that changes in the system that occur in shorter time
intervals than this window may be completely obscured. Thus, the central frequency
should be kept to the smallest value possible to provide the needed frequency resolution
without compromising the ability of the wavelet to detect nonlinear and nonstationary
phenomenon.
4.2.2.5
End Effects: Comparison of Zero Padding and Reflective Padding
A comparison of the proposed reflected padding technique and the zero padding solution
in Torrence & Compo (1998) is shown in Figure 4.17, zooming in at only one end, since
the effects manifest themselves identically at both ends of the signal. The example shown
in Figure 4.17a is the magnitude of the wavelet skeleton for a 2 Hz cosine wave. In this
case the signal begins and terminates with a value of 1. Again, in theory this skeleton
should be a constant value over all time. Without any padding, the skeleton magnitude
gradually approaches the anticipated constant value. Padding with zeros actually
amplifies the characteristic rounding within the cone of influence. On the other hand,
using the padding scheme proposed herein, the wavelet skeleton magnitude achieves a
constant value immediately, mitigating the presence of end effects. The theoretical
magnitude of the wavelet skeleton is shown by a dotted line, which is actually right on
top of the reflective padded result. Figure 4.17b shows the same analysis for a sine wave.
In this case, the skeleton magnitudes without padding are not as dramatically diminished,
128
reflective padding
|W(a=ar,t)|2
|W(a=ar,t)|2
reflective padding
no padding
zero padding
no padding
zero padding
(a)
(b)
t [s]
t [s]
FIGURE 4.17. (a) 2 Hz cosine; (b) 2 Hz sine – influence of padding on
wavelet skeleton magnitude
due to the fact that the signal already begins and ends with zero values, providing some
periodicity of the signal from a Fourier perspective. Using the odd function, reflective
padding scheme recommended here, the end effects are effectively mitigated. While the
use of zero padding to the nearest power of two may increase the efficiency of
calculations when Wavelet Transforms are calculated via Fourier Transforms, this
measure does nothing to mitigate end effects arising from incomplete information and
may even enhance them. Subsequent examples throughout this study will demonstrate the
improvements in wavelet-based system identification as a result of this padding
operation, as well as the lingering limitations.
4.2.2.6
Tuning Temporal Separation
It should be noted that temporal resolutions of wavelets not only dictates the end effects
phenomenon, but also governs the ability to separate events in close temporal proximity.
Equation 4.8 may be used to determine the central frequency necessary to separate two
129
closely spaced events in time. The actual duration of signal characteristics from
inspection of the time series, e.g. the time duration of a single peak or pulse in the signal,
gives some indication of the time resolution that would be required. When deciding what
central frequency is appropriate for a given analysis, the compromise between time and
frequency resolution must always be carefully considered. The recommendation of this
work is to approach analysis in tiers, particularly for complex systems or for those with
more wideband characteristics. Such tiered analyses are presented in Chapters 5 and 7,
with the examples in Chapter 7 providing more discussion on how to identify the
necessary time resolution from characteristics of a time series. Just like zoom transforms
in Fourier analysis (Bendat & Piersol, 1986), a fine frequency resolution wavelet analysis
can be performed to accurately identify the various components in the signal. Then over
each identified range of frequencies, a more temporally refined Wavelet Transform may
be conducted to unveil time-varying properties, as illustrated later in Chapter 7.
4.3 Discretization of Time-Frequency Plane
The use of this continuous Wavelet Transform is attractive not only for its analogs to
Fourier Transforms, but more importantly since it is not constrained in terms of
resolution by the rigid discretization of Discrete Wavelet Transforms. The use of
continuous Wavelet Transforms permits finer frequency resolution necessary for the
nature of the analysis of the narrowband signals encountered in this research. As
discussed in Chapter 3, frame theory governs the discretization of the time-frequency
plane for windowed Fourier Transforms and Wavelet Transforms. In the case of the
Morlet wavelet introduced in this chapter, its intimate relationship to the Fourier
130
Transform implies that it shares many aspects of the frame theory associated with
windowed Fourier Transforms. Work by Daubechies (1990), discussed in Carmona et al.
(1997) and Mallat (1998), identified a critical sampling density called the Nyquist density.
In this framework, given that the time axis is sampled at increments of δt and the
frequency axis at δf, a complete representation is obtained when
δtδf = 1 .
(4.20)
According to the Balian-Low Theorem, this would require the window function
on a Fourier basis to be either non-smooth or have slow time decay. This violates the
basic conditions placed on the parent wavelets in Chapter 3 and excludes many windows
with desirable properties used in Short-Time Fourier Transforms. As a result, an
orthogonal Fourier basis with a differentiable window of compact support, either through
Morlet wavelets or Short-Time Fourier Transforms, is not possible. (Note this fact
motivated largely the popularity of many Discrete Wavelet Transforms.) The choice of a
discontinuous window, e.g rectangular (boxcar), yields an orthogonal windowed Fourier
bases according to the condition in Equation 4.20. Though this window is commonly
used in Fourier analysis, its utility in Short-Time Fourier Transforms is limited due to its
bad frequency localization. In general, all windows satisfying the expression in Equation
4.20 will be poorly localized or have poor regularity (Carmona et al., 1998).
One alternative would be to undersample the space by having δfδt > 1, yielding a
representation that is not complete. Or conversely, sampling at less that 1 to yield a
representation that may be overcomplete (Carmona et al., 1998). In the development of a
131
disccretization of scheme in this work for the Morlet wavelet, the following observations
become relevant:
•
The representation can still be complete as long as the full time-frequency domain
is covered by the Heisenberg boxes of the sampled times and scales, forcing the
sampling at less than the Nyquist density.
•
In reconstruction, there are stability concerns associated with Wavelet Transforms
in such cases, though that is not an issue, given that reconstruction is not within
the scope of this work.
•
A complete but oversampled representation may be redundant and thus strategies
to minimize redundancy must be formulated.
Unfortunately, few authors give much thought to the appropriateness of
discretizations used in conjunction with Morlet wavelets. At minimum, the discretization
of the time-frequency domain must be less than the Nyquist density to avoid the loss of
information, without resorting to a resolution that is so fine that it results in a prohibitive
redundancy of information. Torrence & Compo (1998) simply proposed the use of
fractional powers of two to discretize the scale (frequency) variable. However, this does
not account in any way for the bandwidth of the parent wavelet, potentially providing
redundant information in the frequency domain, even if the Nyqusit density condition is
satisfied. Other applications may arbitrarily or uniformly discretize the analysis scales,
with no regard for frame theory and the frequency resolution of the parent wavelet.
Therefore, this study will address this issue by exploiting the flexibility this continuous
132
transform affords through an adjustable discretization framework that is considerate of
the wavelet bandwidth.
The duration of the Morlet wavelet in the frequency domain based on the meansquare definition of Gabor in Equation 3.6 can be used as a guideline for the specification
for the discrete frequencies at which to evaluate the transform to minimize the
redundancy and gain meaningful information. Assuming the retention of all sampled time
domain data forces δt in Equation 4.20 to the inverse of the signal sampling rate fs. Thus
the discretization of scales to insure sampling at less than the Nyquist density requires
only that δf<fs.
For the Morlet wavelet, the basic frequency bandwidth δf is given by 2∆fi, where
∆fi is defined in Equation 4.5. This requires that
2 ∆f g f i
fo
< fs
(4.21)
The maximum frequency fi that can be considered in an analysis is the Nyquist frequency,
or one-half the sampling rate. As a result
∆f g < f o
(4.22)
where ∆fg, given in Equation 4.4, takes on a value of approximately 0.113 Hz. Recall that
to satisfy admissibility conditions, fo must take on values greater than 0.8, thus satisfying
this condition. Therefore, for all practical applications, as long as the scales are
discretized so that adjacent frequencies are separated by one bandwidth of the analyzing
133
wavelet, the sampling will be less than the Nyquist density and will adequately represent
the signal’s time-frequency domain.
The above discussion demonstrates that the wavelet representations meeting these
minimum requirements are complete, however, the issue of overresolution must still be
addressed. Figure 4.18 demonstrates this concern. The figure shows four slices of wavelet
scalograms at a given time, yielding wavelet instantaneous frequency spectra. The signal
analyzed is a sine wave at 1 Hz. The scales of the Wavelet Transform are discretized by
an arbitrary spacing, inconsiderate of the bandwidth. As the discretization of scales
becomes finer and finer, the bandwidth of the wavelet analysis windows at adjacent
frequencies begins to overlap. By allowing significant overlap between wavelet analysis
windows at neighboring frequencies, significant overresolution is apparent, particularly
in the last spectrum. Also note that the use of arbitrary discretization that does not vary
with frequency causes the lower frequencies to be overresolved more significantly than
higher frequencies.
To avoid this, a framework for the discretization of scales is presented here,
attentive to the frequency-dependent bandwidth of the Morlet wavelet. The frequencies to
be evaluated in the wavelet analysis are specified by
~
f = [ f 1 ≡ f max , f 2 = f 1 − OF∆f 1 ,K, f i = f i −1 + OF∆f i −1 ,K, f min ]
(4.23)
where fmax ≤ fs/2 is the maximum frequency of interest, fmin is the minimum frequency of
interest that can be no smaller than 1/T where T is the duration of the analyzed signal, and
OF is introduced as an overlap factor to permit flexibility. Beginning at the highest
desired frequency and working downward insures that the discretization is conservative,
134
∆fi
∆fi-1
fi-1
∆a=0.1
∆fi+1
fi
fi+1
fi-1
∆a=0.5
∆a=0.25
∆fi-1 ∆fi
fi
∆fi+1
fi+1
∆a=1
FIGURE 4.18. Implications of overlapping analyzing frequencies
as the bandwidth at higher frequencies is larger. The vector of frequencies is then
transformed, according to Equation 4.3, to scales for analysis. The overlap factor, when
set to 2, discretizes at the bandwidth of the Morlet wavelet, in a mean-square sense,
satisfying the Nyquist density condition as long as the central frequency satisfies
Equation 4.22. Increasing the overlap factor beyond 2, provides a more conservative and
potentially incomplete representation if
OF∆f g
2
< fo
(4.24)
is not satisfied. Thus overlap factors of 2 and under are used in this study to provide a
smooth description of frequencies in the wavelet analyses and insure that Equation 4.24 is
satisfied.
135
4.4 Ridge Extraction Techniques
When the Fourier Transform of the parent wavelet is sharply concentrated at a fixed
value of frequency, as is the case of the Morlet wavelet in Equation 4.2, the continuous
Wavelet Transform will have the tendency to “concentrate” at the frequency values
associated with dominant harmonics in the signal, defining a series of curves called
ridges that evolve with time. These ridges, discussed in Section 3.7.3, are locations where
the frequency of the scaled wavelet coincides with the local frequency of the signal, as
shown in Equation 3.30. Note that this equation illustrates that the scales corresponding
to these ridges can be directly used to identify the instantaneous frequency, as shown in
Figure 4.19. The ridge in the time-frequency domain is shown, as well as a slice of the
scalogram at a specific time to yield an instantaneous spectrum focused at the
instantaneous frequency of the system and with measurable bandwidth indicative of the
richness of frequency content about this mean frequency component. While this avoids
the need for the differentiation in Equation 3.35, the wavelet phase can still be used, often
more precisely, to determine the instantaneous frequency for the class of asymptotic
signals, using wavelet skeletons introduced in Chapter 3. The extracted skeletons are also
shown in Figure 4.19.
The extraction of the ridges and their associated skeletons can be accomplished by
a number of techniques. The differential methods are perhaps the most basic way to
detect the ridge. In the case where the signal possesses a single ridge, it is sufficient to
seek the maxima of the wavelet modulus over the scale variable at every time instant.
This global maxima search is defined as
136
|W(a,t=ti)|2
Re[|W(a=ai,t)|],
Imag[|W(a=ai,t)|]
t [s]
|W(a,t)|2
f [Hz]
Ridge of
Transform
f [Hz]
t [s]
WIFS: 2D View of Ridge in TimeFrequency Plane
f [Hz]
f [Hz]
WT: 2D View of Time-Frequency
Plane
t [s]
t [s]
FIGURE 4.19. Schematic of various time-frequency characteristics of
Wavelet Transforms
W (ar , t ) = max W (a, t )
a
.
(4.25)
The ridge is then given by a graph of the function ar(t) (Carmona et al., 1998). While this
will yield a ridge estimate for the most rudimentary applications, it becomes evident that
this approach will only be capable of capturing one ridge and must be modified to
identify multiple local maxima for multicomponent signals. This modification is
achieved, for example, by identifying peaks in the wavelet modulus corresponding to the
location of ridges
∂ W ( a, t )
∂a
= 0.
a = ar ( t )
137
(4.26)
Unfortunately, this technique is less robust in the presence of noise, which introduces
additional local maxima (Carmona et al., 1998). More precision can be achieved using
the Marseille Method based on the wavelet phase. These methods identify the locations
where the wavelet modulus shows a high degree of similitude with frequency
components within a signal.
The ridge extraction approach can be an even more powerful tool when it is
manipulated to extract an asymptotic signal from noise. Unlike the differential
approaches, this more sophisticated class of techniques exploits the fact that the
asymptotic signal has a ridge that is a smooth function of time, unlike the ridges
associated with noise or other intermittent signal components that may appear as
scattered fragments of a curve throughout the time-frequency plane. In addition to using
the smoothness of the ridge to aid in the extraction scheme, information on the
characteristics of the noise may also be integrated into the search algorithm to enhance
performance.
Using the fact that the wavelets localize their energy near ridges and that the
ridges of the asymptotic signals are smooth and slowly varying functions, a penalty
function can be formulated
ε (a r ) = − ∫ SG (a r (t ), t )dt + λ ∫ a r′ (t ) dt + µ ∫ a r′′(t ) dt
2
2
(4.27)
and the ridge detection problem is recast as a minimization problem on this function, as
discussed in Carmona et al. (1998). A Bayesian approach as well as other techniques can
be used to propose penalty functions for seeking out smooth ridges. Since the numerical
algorithms used to solve for the ridge in this manner can be unstable, alternative
138
approaches are provided in Carmona et al. (1998), including the “Corona” Method and
the Iterated Conditional Modes (ICM) technique. These approaches are very attractive for
separating a single asymptotic signal ridge from a bed of noise, but if there are multiple
asymptotic ridges, more robust algorithms are required. This approach does not use any
type of penalty function minimization, but rather introduces a stochastic system to
generate occupation densities that draw the ridges through the Crazy Climber Algorithm
(Carmona et al., 1998).
It should also be noted that the phase of the WT could also be used to isolate the
ridges. As will be demonstrated in subsequent examples, the instantaneous frequency
identified by the ridge is often more precise, but also more sensitive to noise (Carmona et
al., 1997). Feldman & Braun (1995) noted that the estimate of the instantaneous
frequency from phase information may be high in variance and concluded that a lower
variance estimate can be obtained directly from the maxima of time-frequency
distributions, an observation previously confirmed by Boashash (1992b). This is an
important consideration for techniques based on Hilbert Transforms, which rely entirely
on phase information to define the instantaneous frequency. Thus, the flexibility of dual
venues for the instantaneous frequency estimation is a valuable asset for the Wavelet
Transform.
Though the extraction techniques geared specifically to separate asymptotic
signals are powerful and attractive, smooth ridges are often not present in the case of
arbitrary natural signals or random processes. Therefore, for the more general analyses of
signals throughout this study, the more traditional differential extraction methods
139
(Equation 4.26) are employed so that ridges from a variety of signal types and sources
can be isolated. Further, the ridges identified in this manner are plotted as colored
contours in the time-frequency plane, preserving the magnitude of the various ridge
components. This perspective is termed by the author as a wavelet instantaneous
frequency spectrum (WIFS)
⎧⎪SG (a, t ) a = a (t )
r
WIFS (a, t ) = ⎨
.
⎪⎩ 0 a ≠ ar (t )
(4.28)
In this way, the dominant ridges can be differentiated from fainter ridge components that
are often artifacts of noise variations or “shadows” of the dominant ridge, created by the
side lobes of parent wavelet, G(ω). An example of the WIFS is also shown in Figure
4.19.
4.5 Conclusions
The multi-resolution capability of Wavelet Transforms makes it a powerful tool for timefrequency analysis, optimally adjusting to the Heisenberg Uncertainty Principle at each
analysis frequency. However, in order to provide meaningful results, the transform must
be applied with a proper understanding of its time-frequency resolutions. As these issues
are rarely discussed within the literature, in part due to the infancy of Wavelet
Transforms, a unified analysis framework is presented herein for the Morlet wavelet. This
wavelet is advocated for use in this study by virtue of its analogs to Fourier Transforms
and Fourier frequencies, making it well-suited for modal analysis familiar to most
practicing engineers. A discussion of the resulting wavelet resolutions reiterated the
importance of optimally adjusting the central frequency of the parent wavelet to achieve
140
the desired resolutions, e.g. to separate closely spaced modes. This discussion also
provided a theoretical basis for the presence of end effects and offered a practical remedy
in the form of reflective padding to extend the length of the time series to mitigate the
influence of end effects. The chapter concluded by establishing a flexible framework for
discretizing the time-frequency plane, cognizant of the bandwidth of the Morlet wavelet,
and summarized strategies for ridge extraction, introducing the notion of wavelet
instantaneous frequency spectra representative of the dominant frequency components
within the signal. This processing framework provides the basis for all wavelet analyses
that follow in the next four chapters.
141
CHAPTER 5
WAVELETS FOR SYSTEM IDENTIFICATION, PART I:
APPLICATIONS TO WIND, WAVE AND SEISMIC ANALYSIS
5.1 Introduction
Civil Engineering structures assume a wide array of structural forms and dynamical
properties, as shown in Figure 5.1, subjecting the various classes of bridges, towers,
buildings and ocean platforms to the threats of wind, wave and earthquake loads. These
diverse natural loadings present further challenges in that they cannot be completely
captured in a stationary analysis framework such as that provided by Fourier Transforms.
As motivated earlier in Chapter 3 and in work by Gurley and Kareem (1999), wavelets
provide a new venue for the detection of nonstationary features in natural loading and the
resultant response previously obscured by traditional spectral representations. This
potential is more fully explored in this chapter by applying a continuous wavelet analysis
to measured wind, wave and seismic signals serving as input to Civil Engineering
structures. The analyses also consider measured full-scale response of buildings and
experimental investigations of offshore platforms under wind and wave action,
representing the characteristics of each through wavelet scalograms, wavelet
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Reduced Spectrum
FIGURE 5.1. Spectra of environmental loads acting on
Civil Engineering structures (taken from Kareem, 1987)
instantaneous frequency spectra and wavelet instantaneous spectra to reveal the
distribution of signal energy with respect to both time and frequency. A comparison of
wavelet marginal spectra and Fourier spectra is also provided to demonstrate the
influence of wavelet multi-resolution characteristics on traditional spectral perspectives.
Before these examples are introduced, however, this chapter will first introduce some
additional wavelet theory and wavelet representations that will be employed herein,
whose physical significance is demonstrated using a simple simulated signal.
5.2 Preliminaries
As discussed in Chapter 3, the wavelet scalogram, denoted as SG(a,t) by letting b ≡ t,
was simply defined as the squared modulus of the wavelet coefficients. However, as
143
discussed in Chapter 4 in the context of the wavelet transformed harmonic components,
the coefficients associated with a particular harmonic are actually weighted by the square
root of the scale. Therefore, the amplitude of high frequency components in the
scalogram will always be reduced by a factor of a and be smaller relative to the low
frequency components. This convention arose from the definition of the wavelet
transform according to Mallat (1998), who adopted this L2 normalization in the
development of expressions to fully recover the signal energy in wavelet reconstructions.
Besides its relevance to the conservation of energy and signal reconstruction, the
weighting of the scalogram values by a essentially normalizes them relative to the size of
the frequency window. Recall in Chapter 3 that the duration of the wavelet analysis
window at a given frequency is inversely proportional to scale, implying that a wavelet
coefficient associated with a high frequency was formed from data over a larger
frequency range than its low frequency counterpart. The energy density of the scalogram
can then be viewed as energy divided by the length of the frequency domain window.
While this normalization is sensible considering the multi-resolution character of
wavelets, it implies that a direct comparison of wavelet magnitudes cannot always be
used to distinguish the relative contributions of a frequency component at a given instant
in time, i.e. through instantaneous spectra or WIFS. Instead, the scalogram and WIFS can
be used to identify the presence of particular frequency components in time and the
frequencies with the highest energy densities relative to their respective frequency
resolutions. To uncover the relative contributions of frequency components energy
measures associated with the wavelet scalogram must be consulted.
144
The marginal wavelet power spectrum, also referred to as the mean wavelet
spectrum (Lewalle, 1998), can be estimated by integration over the time variable as
discussed in Perrier et al. (1995), which when implemented for a signal of finite duration
T, may be expressed as
SWT (a ) =
2
Cg
⎡1 T
⎤
⎢ T ∫ SG (a, t )dt ⎥
⎣ 0
⎦
(5.1)
where Cg is the admissibility factor or norm of the parent wavelet, introduced in Chapter
3, and the factor of two is necessary since the analytic wavelets, such as the Morlet,
suppress all negative frequencies. Note that the Fourier frequency corresponding to each
scale in Equation 5.1 can be determined through the inverse relationship, making SWT
equivalently a function of frequency that can be readily compared to traditional Fourier
spectra. In the case of the Morlet Wavelet, the admissibility factor is a function of the
central frequency and can be determined by the numerical integration of the following
expression
T2
G (ω )
T1
ω
Cg = ∫
2
T2
2πe −(ω −ωo )
T1
ω
dω = ∫
2
dω .
(5.2)
Note that the definition of the admissibility condition in Chapter 3 specified limits of
integration from zero to infinity. For this analytic wavelet, however, the localization of
the Morlet wavelet in the frequency domain allows the numerical evaluation of Equation
5.2 to be conducted over a finite range of frequencies. Recall that the mean-square
definition of the Morlet’s frequency resolution, introduced in Chapter 3, spans one
standard deviation of the Gaussian window. Extending this notion, 99.7% of the Gaussian
window density lies within 3 standard deviations of the mean value, which in this case is
145
the central frequency. Thus, the infinite limits of integration can be reduced to just
beyond the effective duration of the Gaussian window itself, to ease computation.
Therefore, let T1 and T2 be defined as
T1 = ω o − 5(∆ω ) = ω o − 5⎛⎜ 1 ⎞⎟ and T2 = ω o + 5(∆ω ) = ω o + 5⎛⎜ 1 ⎞⎟ .
2⎠
2⎠
⎝
⎝
(5.3)
Note that a parametric investigation affirmed that assuming five standard deviations was
sufficient to accurately compute the admissibility factor, as visualized when plotting the
function G(ω)/ω in Figure 5.2 for the case of a ωo=4π rad/s Morlet wavelet.
|G(ω)|2/ω
5σ
5σ
ωo
ω [rad/s]
FIGURE 5.2. Admissibility function for 2 Hz
Morlet wavelet
146
The mean square value or variance of the assumed zero mean signal can be then
determined from the wavelet marginal power spectrum by integrating over the range of
scales
an
σ WT 2 = ∫ SWT (a)
a1
da
.
a2
(5.4)
Note that the limits of integration are defined in terms of the finite range of scales
considered in the analysis and will identically capture the signal variance as a1→ 0 and
a2→∞, equivalent to the theoretical expression in Mallat (1998). Similarly, integrating
over the range of Fourier frequencies can also provide an estimate of signal variance, as
discussed in Mallat (1998),
σ WT 2 = −
1
fo
fn
∫S
WT
f1
⎛ f o ⎞df
⎜ f⎟
⎝
⎠
(5.5)
where SWT(fo/f)≡SWT(a), f1=fo/a1 and fn=fo/an. The factor fo emerges from the change of
variables between Equation 5.4 and 5.5, and the negative sign arises from the inverse
relationship between scale and frequency, essentially inverting the limits of integration.
Either expression can be used to obtain a wavelet-based estimate of the signal variance
that is equivalent to the true variance, provided that the limits of integration approach [0,
∞].
The wavelet marginal spectrum will be compared to the traditional Fourier
spectrum in this chapter to highlight the added low-frequency insights afforded by
wavelets. Ideally the Fourier spectra computed via the Fast Fourier Transform and
marginal wavelet spectrum should be calculated using similar frequency resolutions, for a
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fair comparison. However, due to the multi-resolution character of wavelets, this is not
possible. Instead, the resolution of the Fourier analysis is kept close to a wavelet mean
frequency resolution. The wavelet transform is calculated for a range of scales, associated
with Fourier frequencies ranging from zero up to the Nyquist frequency, i.e. fs/2.
Therefore, the wavelet mean frequency resolution, ∆ f is defined as
fWT
∆f =
f o 2π 2
(5.6)
where fWT is the mean of the Fourier frequencies considered in the wavelet analysis. A
mean temporal resolution can be defined in a similar manner
fo
∆t =
2 fWT
.
(5.7)
These two measures will be referenced in some examples within this chapter to indicate
the relative accuracy of the wavelet analysis. The frequency resolution of the Fourier
analysis ∆fFFT is then determined by selecting the number of FFT points (NFFT), to the
nearest power of two, that, when inverted and multiplied by the sampling rate fs, is
comparable to the mean wavelet mean frequency resolution. Fourier spectra are then
calculated by breaking the signal into Ns blocks of length NFFT, spanning T= NFFT / fs
seconds. The Fourier coefficients are calculated over that block of discretely sampled
points xn, via fast Fourier transforms (Bendat & Piersol, 1986), according to
Xk =
1
fs
⎡ 2πkn ⎤
∑ xn exp ⎢− i
⎥
n =0
⎣ N FFT ⎦
N FFT −1
(5.8)
where k is the index corresponding to the discrete frequencies of the Fourier spectrum
148
fk =
k
T
k = 0, 1, 2, …, N/2.
(5.9)
An estimate of the power spectrum is then made by averaging the squared
magnitude of the Fourier coefficients from each block, according to (Bendat & Piersol,
1986)
S FFT ( f k ) =
2
2 Ns
∑ Xk .
N sT i =1
(5.10)
Note the averaging here replaces the expected value operation, and the factor of 2 is
introduced to compensate for the fact that this is a one-sided spectral density, as indicated
in Equation 5.9. The signal variance associated with the Fourier power spectrum is then
estimated by the following expression
fN / 2
σ FFT 2 = ∫ S FFT ( f k )df
(5.11)
f0
In addition to comparisons of the wavelet marginal spectrum and its Fourier
counterpart, wavelet measures of energy accumulation are also presented throughout this
chapter. The energy accumulations in the frequency domain, denoted E(f), are determined
by an integral operation at each frequency of the wavelet marginal spectrum, according to
fi
f
E ( f i ) = ∫ SWT ⎛⎜ o ⎞⎟df
⎝ f⎠
f1
for i=1, 2, .. n
(5.12)
for each of the n Fourier frequencies considered in the wavelet analysis. Similarly, a time
analog to the marginal spectrum may be defined as
149
fn
f
SWT (t ) = − ∫ SG⎛⎜ o , t ⎞⎟df .
⎝ f ⎠
f1
(5.13)
Energy accumulations in the time domain, denoted E(t), are then appropriately
determined by
tj
E (t j ) = ∫ SWT (t )dt
for j=1, 2, .. m
(5.14)
t1
for each of the j discretely sampled time ordinates considered in the wavelet analysis.
Rates of change or derivatives of each of these accumulation measures, with respect to
frequency or time, denoted respectively as dE(f)/df or dE(t)/dt, and the maximum values
of these accumulation rates will be of particular interest in subsequent discussions as they
identify the arrival of significant events in time and frequency.
As mentioned previously, the assessment of the relative strength of a frequency
component at a given instant in time should not be based on the amplitude of wavelet
coefficients, but rather by the energy associated with that scale and can be determined by
integrating the area under an instantaneous spectral peak. For this reason, a relative
energy measure is provided by this work. Instantaneous spectra are merely scalogram
coefficients associated with a particular instant in time that peak at each instantaneous
frequency component denoted by IFi. Therefore, the relative contribution to the
instantaneous signal energy at that time tj from that frequency component is defined here
as the ratio of the area under that particular spectral peak to the overall area under the
instantaneous spectrum
150
f 2 ,i
Eˆ i (t = t j ) =
⎛
∫ SG⎜
⎝
f1,i
fo
f
, t = ti ⎞⎟df
⎠
⎛f
⎞
∫ SG⎜ o f , t = ti ⎟df
⎝
⎠
f1
fn
.
(5.15)
In this case, the limits of integration in the numerator are the frequencies associated with
the initiation and termination of an individual spectral component. Note each of the
integrations specified in Equations 5.1, 5.2, 5.4, 5.5 and 5.11-5.15 are achieved
numerically in this chapter using trapezoidal integration.
The following simulated example will demonstrate the utility of these additional
wavelet spectral representations and energy measures, noting their ability to identify the
arrival and participation of different frequency components. Armed with an
understanding of these measures and wavelet visualizations, the analysis approach
introduced here will then be extended to a number of measured earthquake ground
motions and structural responses to earthquakes, winds and waves.
5.3 Example
The physical significance of the preceding measures is demonstrated by a simplified
example. Consider a 100 second signal sampled at 10 Hz. For the first 50 s of the signal,
a lone cosine described by 5cos(8πt) is present. In the last 50 s of the signal, this is
supplemented by an additional cosine of the same amplitude and half the frequency. A
Morlet wavelet analysis with fo = 3 Hz is conducted. This produces a wavelet mean
frequency resolution of 0.103 Hz and mean temporal resolution of 0.771 s. Accordingly,
128 FFT points were selected to produce Fourier spectra with a resolution of 0.078 Hz.
151
The wavelet padding scheme discussed in Chapter 4 was implemented for β = 3 and an
overlap factor, OF =1 was selected to produce the smooth wavelet spectra with minimal
computational effort. As noted by Gurley & Kareem (1999), discretization of scales can
be critical in insuring that a wavelet analysis accurately captures signal energy and
spectral shape, often limiting the utility of the DWT in these arenas. The flexibility
provided by the overlap factor, allows finer discretizations at the expense of
computational effort. As shown in Table 5.1, the OF = 2 provides the most proper
discretization of scales in terms of bandwidth definitions discussed in Chapter 4 and
marks the maximum allowable OF providing reasonable estimates of signal variance. By
increasing OF and essentially over resolving the space, there is no marked improvement
in results despite greater computational expense. Thus the ability of the wavelet
transformed data to replicate the actual RMS of the signal serves as a metric to assess the
quality of the analysis parameters chosen.
TABLE 5.1
INFLUENCE OF OVERLAP FACTOR ON WAVELET ENERGY ESTIMATES
OF
4
3
2
1
0.5
Error in Energy Estimate
-13.4%
11.2%
0.345%
-0.140%
-0.162%
The resulting scalogram and wavelet instantaneous frequency spectrum are
provided in Figure 5.3. Note that both the scalogram and the wavelet instantaneous
152
SWT(fo/f), SFFT(f)
x(t)
f [Hz]
f [Hz]
f [Hz]
t [s]
FIGURE 5.3. Example signal, wavelet scalogram and WIFS (left, top to
bottom) and marginal spectral comparison with Fourier power spectrum
(right)
frequency spectrum isolate the involved frequency bands as well as the time at which the
second frequency contribution appears, albeit more precisely in the form of ridges in the
WIFS. The relative magnitude of the peaks in the Fourier spectrum identify the higher
frequency component as the more dominant in the signal, but this does not give any
indication of how long each component was present. The Fourier spectrum could be
interpreted as two frequency components of equal amplitude, but with the low frequency
component only present a fraction of the time (as is the case here) or as two frequency
components both present for the same amount of time, but the higher frequency
component having a larger magnitude. The wavelet marginal spectral amplitudes are
much less than their Fourier counterpart, due to the increased bandwidth from the
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Gaussian window. In order for energy to be conserved by the wavelet, these amplitudes
must be smaller. The relative magnitude of the wavelet peaks, if interpreted in the same
manner as the Fourier spectrum, would indicate that the low frequency component
slightly dominates. Recall however that the bandwidth of the high frequency component
is larger than the low frequency component, as a result of the frequency dilation of the
wavelet analysis window. The larger bandwidth implies that a lower spectral amplitude is
necessary to insure that the area under the spectral peak corresponds to the true energy
contribution from that particular frequency. This emphasizes the need to consider the area
under the wavelet spectrum rather than its spectral amplitude when assessing the relative
contributions of two well-separated components. Still, the overall signal energy is well
captured by the wavelet. The RMS value of the signal is 4.34, while the area under the
wavelet marginal spectrum produces a value of 4.33, and the area under the Fourier
spectrum is 4.25. This proves that the measure in Equation 5.5 does compensate
appropriately for the multi-resolution character of wavelets, which results in the modified
spectral amplitudes.
The energy accumulation plots are provided in Figure 5.4 and are useful in
determining critical events within the course of a signal. As the frequency energy
accumulation plot at the left demonstrates, when read from right to left, the signal has no
energy until 4 Hz, where it receives approximately two thirds of the total signal energy.
The plateau between 4 Hz and 2 Hz further indicates that there are no additional energy
contributions in this range. The increase to 100% energy at 2 Hz affirms that the
remainder of the signal energy resides at this frequency. In time, it is similarly affirmed
that one third of the signal energy is present at 50 s, when the second cosine is
154
E(t) [%]
E(f) [%]
f [Hz]
t [s]
FIGURE 5.4. Energy accumulation in frequency
domain (left) and time domain
introduced, changing the slope of the energy accumulation. The rates of change of the
accumulation plots, presented in Figure 5.5, identify the arrival of specific events in time
and frequency. The maximum change in energy in the frequency domain occurs at 4.05
Hz, corresponding to the dominant frequency component in the signal. The dramatic shift
in the rate of energy accumulation in time is evident in the left image, with the maximum
energy in the time domain being fully realized at 53 s and sustaining itself for the
duration of the signal. Understandably, the major changes in energy accumulation with
frequency occur at the two harmonics of the signal. Though for this simple example, the
plots in Figure 5.5 don’t seem to provide any significant additional information, these
measures have greater utility in the case of more complex signals with a richer
distribution of frequencies and nonstationary characteristics, which will be considered
155
SWT(fo/f)
x(t)
dE(t)/dt
dE(f)/df
f [Hz]
t [s]
FIGURE 5.5. Signal and rate of change of energy accumulation in time
domain (left) and marginal spectrum and rate of change of energy
accumulation in frequency domain for example cosine signal
subsequently. Note again that these energy accumulations, being based on the area under
the wavelet spectra, again compensate for the amplitude distortions that result from the
multi-resolution analysis and therefore provide a true representation of the energy
contributed by each frequency component.
The final series of plots in Figure 5.6 demonstrates the instantaneous spectra at 4
distinct time intervals in the course of the signal. The first plot is taken from the signal
when only the 4 Hz component is present. The subsequent pair of plots show the
appearance of the second component in the vicinity of 50 s, as the energy associated with
that component gradually phases in to the scalogram. Once both components are fully
present in the last plot, it is evident that the magnitudes of the instantaneous spectral
peaks are not equal, despite the two cosines having the same amplitude. Again, this is to
insure energy conservation, since the higher frequency peak is associated with a larger
frequency range and thereby greater bandwidth, accounted for by the scaling of wavelet
156
SG(a,t=48.5s)
SG(a,t=20s)
SG(a,t=51s)
SG(a,t=75s)
f [Hz]
f [Hz]
FIGURE 5.6. Wavelet instantaneous spectra taken at critical time steps
in the example signal evolution: t=20 s, 48.5 s, 51 s, 75 s
scalogram values by a. However, by integrating the areas under the spectral peaks in each
plot of Figure 5.6 and dividing by the total area under the curve as described in Equation
5.15, the relative strength of each component at that instant in time is assessed. The
results are tabulated in Table 5.2 and indicate that the individual frequency components
were identified with sufficient accuracy and their relative contributions, outside of the
transitory period, were accurately identified, again affirming that by considering spectral
energy and not amplitude, the effects of a multi-resolution analysis can be adequately
compensated for. Note that the transitory period exists since the wavelet considers
information to the past and future of a given time instant. Therefore, the occurrence of the
157
TABLE 5.2
RELATIVE CONTRIBUTIONS OF EACH COMPONENT TO INSTANTANEOUS
SPECTRA FOR EXAMPLE
tj
IF1
Eˆ1 (t j )
IF2
Eˆ1 (t = t j )
20 s
49 s
51 s
75 s
n/a
1.98
1.98
1.98
n/a
7.4%
45%
50%
3.98
3.98
3.98
3.98
100%
92.3%
54.7%
50%
second cosine is not pinpointed at an instant, but its occurrence is sufficiently localized in
time. The scalograms, wavelet instantaneous frequency spectra and wavelet marginal
spectra are considered throughout the subsequent discussions to permit a general
identification of frequency components in time and to isolate meaningful events. The
other wavelet energy measures based on wavelet instantaneous spectra are then consulted
to determine the relative contributions of each frequency component or event in the
signal.
5.4 Earthquakes
The birth of wavelets can actually be traced back to its applications in seismology. It was
Morlet’s work in reflection seismology that led him to propose sending shorter
waveforms at high frequencies obtained by scaling a single function called a wavelet. At
the same time, Grossman was working on coherent quantum states and realized the
applicability of Morlet’s idea to his field. The collaboration between these two led to the
formalization of the continuous wavelet transform, known in French as Ondelettes
158
(Grossman & Morlet, 1984). Though these ideas were not new to mathematicians
working in harmonic analysis and researchers in multi-scale image processing, the work
of Grossman and Morlet, as well as the mathematician Meyer, brought together
researchers from a variety of fields to unify the theory of wavelets. Therefore it is only
fitting to return to the seismic problem as the first example of wavelet applications in
Civil Engineering. As the discussions in Gurley & Kareem (1999) suggest, there is
considerable promise in the applications of wavelet transforms in the analysis of not only
seismic ground motions, but also the resulting response. The following sections apply the
aforementioned wavelet measures in the analysis of nearfield records from five seismic
events: El Centro (1940), Mexico City (1985), Loma Prieta (1989), Northridge (1994)
and Kobe (1995). For these events, only the ground motions associated with the larger of
the two horizontal directions are analyzed – the vertical components are not considered at
this time. For each of these events, due to the short duration characteristics, a fo = 1 Hz
Morlet wavelet was first applied, and a more detailed examination of the frequency
content was achieved using fo = 3 Hz, with overlap factor (OF) of 1. These two analyses
are referred to throughout the examples in this section as the primary and secondary
analyses, respectively. Each signal is analyzed at scales corresponding to Fourier
frequencies from 0.05 Hz to the Nyquist frequency. Dependent on the length of the
record and the central frequency chosen, padding was provided wherever possible using
β = 2 or 3. Subsequently, the analysis of a low-rise structure’s response to the Northridge
quake will also be investigated. It should be cautioned when viewing the following
results that each of these earthquake records and their associated time and frequency
characteristics depend strongly on local soil conditions and topography and should not be
159
considered representative of the complete suite of seismic motions associated with the
event.
Finally, it should be again noted that all wavelet scalograms, marginal spectra,
and instantaneous spectra do not correct for reduced spectral amplitude or enhanced
spectral bandwidth that results from the multi-resolution analysis, though the area under
each spectral peak does appropriately reflect the energy contributions of each component.
For this reason, all energy accumulations and energy measures defined in this chapter and
calculated from the data presented herein are based upon the area under the wavelet
spectra, as discussed previously, and thus directly account for the consequences of multiresolution analyses, providing a true energy measure and gauge of the relative
contributions of each component, which cannot be ascertained directly from wavelet
spectral amplitudes. Therefore, only the plots of energy accumulation and rates of change
of energy accumulation appropriately account for this multi-resolution effect.
5.4.1 El Centro (1940)
The El Centro earthquake struck southern California on May 18, 1940 at 20:37 local
time. The recorded time history of the event, sampled at 50 Hz, is subjected to two
aforementioned wavelet analyses. The mean wavelet resolutions are given for each
analysis in Table 5.3.
Figure 5.7 shows the wavelet scalograms for these two analyses and their
accompanying wavelet instantaneous frequency spectra. The impact of wavelet
resolutions is immediately obvious, as the coarser frequency resolution of the primary
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TABLE 5.3
RESOLUTIONS OF FOURIER AND WAVELET ANALYSES
El Centro
Guerrero MC
Loma Prieta
Northridge
Kobe
fo = 1 Hz
∆fFFT
∆f
0.19 Hz
0.31 Hz
0.19 Hz
0.19 Hz
0.10 Hz
1.41 Hz
1.13 Hz
0.71 Hz
0.71 Hz
0.71 Hz
PRIMARY ANALYSIS
fo = 3 Hz
∆t
∆f
0.05 s
0.07 s
0.11 s
0.11 s
0.11 s
0.47 Hz
0.38 Hz
0.23 Hz
0.23 Hz
0.23 Hz
∆t
0.17 s
0.21 s
0.34 s
0.34 s
0.34 s
f [Hz]
f [Hz]
f [Hz]
f [Hz]
ag
[cm/s2]
ag
[cm/s2]
SECONDARY ANALYSIS
t [s]
t [s]
FIGURE 5.7. El Centro ground motion, wavelet scalogram and WIFS (top to
bottom) for primary (left) and secondary analyses
161
analysis leads to long vertical bands and the coarser temporal resolution of the secondary
analysis produces horizontal energy bands. However, viewing the two analyses in tandem
can be beneficial. The primary analysis provides better temporal information and affirms
that the energy lies dominantly between 4 and 7 s, though mostly near the former. In fact,
three distinct events are detected in the vicinity of 3.6, 6.5 and 13.8 s. The first event has
a broadband energy range the sweeps from 1 to 4 Hz, though focused between 1-2 Hz
until the fifth second, when it narrows to the 1 Hz range. The second event also clusters
in the 1-2 Hz range, while the third event is decidedly in the 1 Hz vicinity. Despite the
presence of low frequency components in the wake of the first event, their contributions
appear from the scalogram to be minor, as are the trace high frequency components. The
WIFS affirms the richness of frequencies particularly associated with the first major pulse
of the quake and to some extent, the secondary pulse, while isolating the persistent 1 Hz
component. A wealth of intermittent high frequency and low frequency contributions is
apparent, specifically in the vicinity of the first event near 4 s.
The secondary analysis separates distinct bands of energy near 1, 1.5, 1.7 and 2
Hz, though clearly concentrated near 1.5 Hz. In agreement with the primary analysis, the
broadband features and dominant energy are localized near 5 s, which extends into the
higher frequencies. The additional burst of energy at 27 s is more clearly associated with
2 Hz. The enhanced resolution of the WIFS in the secondary analysis affirms the richness
of energy at 2.3 Hz and even down to 0.4 Hz. While the presence of contributions
beneath 1 Hz were noted for this record by Huang et al. (1998), their contributions to the
overall signal energy, relative to the other participating frequencies will be addressed
162
subsequently. Note also the variation in the ridges that indicate subtle shifts to higher and
lower frequencies, only evident due to the enhanced frequency resolution.
A comparison of the Fourier power spectrum and the marginal wavelet spectrum
for each analysis is provided in Figure 5.8. Table 5.3 lists the frequency resolution of the
Fourier analysis and the mean frequency and time resolutions of the wavelet analysis.
The traditional Fourier spectrum indicates that there is noteworthy energy below 1 Hz in
this quake, though the vast majority of the signal’s energy concentrates between 1 and 2.2
Hz, with peaks at 1.4 and 2.0 Hz and an additional peak near 4 Hz. Beyond 5 Hz, there is
little energy content within the record. The low frequency energy rapidly falls off beneath
0.84 Hz. As expected the marginal wavelet spectrum in the primary analysis identifies the
same general frequency content and dominance of the 1-2 Hz content, but with less of the
details, due to the lower frequency resolution. The enhanced resolution of wavelets at the
lower frequencies more clearly identifies the distinct presence of a low frequency
contributor between 0.3 and 0.6 Hz, but lacking sufficient energy. The more refined
resolutions in the secondary analysis provide greater detail on the robust distribution of
energy between 0.82 and 2.5 Hz. The energy falls of quickly beyond this range. While a
low frequency component is noted down to 0.4 Hz, it relative contributions again seem to
pale in comparison to the energetic components in the vicinity of 1-2 Hz. While the
general trends in the secondary wavelet marginal spectrum are consistent with that of the
Fourier spectrum, there is a difference in amplitude, which is merely the function of the
finer bandwidth in the secondary wavelet analysis, directly accounted for in the scaling
by a. Still the energy is completely conserved in the wavelet representation, as evidenced
by Table 5.4 when compared to the signal standard deviation σ.
163
PRIMARY ANALYSIS
SWT(fo/f), SFFT(f)
SWT(fo/f), SFFT(f)
SECONDARY ANALYSIS
f [Hz]
f [Hz]
FIGURE 5.8. Wavelet marginal spectra with Fourier power spectrum for
primary (left) and secondary analyses for El Centro quake
TABLE 5.4
SIGNAL STANDARD DEVIATION AS ESTIMATED FROM FOURIER AND
WAVELET MARGINAL SPECTRA
σ
[cm/s2]
El Centro
Guerrero MC
Loma Prieta
Northridge
Kobe
44.06
24.20
71.32
69.06
115.19
σFFT
σWT
[cm/s2]
44.15
24.81
75.44
71.43
155.19
[cm/s2]
fo = 1 Hz
44.10
24.21
71.39
69.12
115.31
fo = 3 Hz
44.07
24.20
71.32
69.07
115.17
The energy accumulation plots account for the multi-resolution characteristics of
wavelets and provide a more objective tool for investigating the dominant energy
components in the signal in comparison to the magnitude of wavelet scalograms, as
shown in Figure 5.9. The primary energy accumulation in time demonstrates a jaggedness
indicative of the arrival of distinct energetic pulses. By compromising this temporal
164
SECONDARY ANALYSIS
f [Hz]
E(t)
E(f)
E(t)
E(f)
PRIMARY ANALYSIS
t [s]
f [Hz]
t [s]
FIGURE 5.9. Energy accumulation in frequency domain and time domain
for primary (left) and secondary analysis of El Centro
PRIMARY ANALYSIS
dE(f)/df
dE(f)/df
SWT(fo/f)
SWT(fo/f)
SECONDARY ANALYSIS
f [Hz]
f [Hz]
FIGURE 5.10. Wavelet marginal spectrum and rate of change of energy
accumulation in frequency domain for primary (left) and secondary analysis
of El Centro
165
resolution in the secondary analysis, a more precise measure of the energy accumulation
in frequency is achieved. To enhance the discussion further, the rates of change of these
energy measures in both frequency and time, are presented in Figures 5.10 and 5.11.
SECONDARY ANALYSIS
dE(t)/dt
dE(t)/dt
ag
ag
PRIMARY ANALYSIS
t [s]
t [s]
FIGURE 5.11. El Centro ground acceleration and rate of change of energy
accumulation in time domain for primary (left) and secondary analysis
Looking first at the accumulation of energy in the frequency domain, from Figure
5.10, it is evident that the maximal rate of change in the primary analysis occurs at 1.35
Hz, though there is limited detail due to the loss of frequency resolution. The frequencies
associated with these elevated rates of energy accumulation span a wide band from 1 to
1.9 Hz, with a minor change in energy associated with 0.4 Hz and no significant energy
beyond 10 Hz. The secondary analysis, by virtue of a refined frequency resolution,
identifies several distinct frequencies at which energy is contributed. Still the, maximum
rate of change in the signal is constrained between 1.15 and 2.0 Hz. Beyond 2.5 Hz, there
is little change in the energy accumulation within the signal. The steep ascent at 2.5 Hz in
166
Figure 5.9, marked by the dotted vertical line in the secondary analysis, affirms this
concentration. Based on the noteworthy frequencies identified in Figure 5.10, an
indication of the energy associated with these frequencies can be determined from the
accumulation plots in 5.9. Dotted lines are used to indicate frequencies of interest.
In the primary analysis, the dominant influx of energy came from the frequencies
between 1 and 1.9 Hz. At this time, the signal energy jumps from 61% to 88%.
Investigating this in more detail through the secondary analysis, it is shown that 37.6% of
the energy is associated with frequencies beyond 2.5 Hz. About 40% of the total signal
energy is associated with frequencies between 1.15 and 2.5 Hz. Frequencies from 0.8 to
1.15 Hz account for only a few percent of the signal energy. Only 8% of the energy lies
below 0.8 Hz. It is clear from the energy accumulation plot in Figure 5.9 that the single
greatest jump in energy is associated with 1.15 Hz, where the cumulative energy rises
from nearly 80% to 90%. From this analysis, it is evident that the range of frequencies
from 1.15 to 2 Hz is the most energetic.
In the time domain, it is evident from Figure 5.11 that the primary analysis is
superior in capturing many of the individual pulses in the record, though both analyses
capture the same overall trends. The primary analysis indicates that most rapid changes in
the energy within the signal occur between 4 and 7 s. Within this time frame, three
distinct events are discernable: 4.1 s, 5.5 s and 6.7 s, with the most rapid change in energy
being associated with the first in this series. The energy falls of very quickly following
this flurry of activity. Beyond 10 s, there are 4 events, all bringing comparable levels of
167
energy: 11.1 s, 13.7 s, 26.2 s and 27.9 s. The occurrence of each of these events is
denoted by dotted lines in the primary energy accumulation plot in Figure 5.9.
The accumulation of energy in time is not smooth due to the arrival of many small
pulses associated with the arrival of the primary (P) and secondary (S) waves. With the
arrival of the first pulse at 4.1 s, the energy accumulated doubles from 14% to 28%. The
intermediate event at 5.5 s brings the energy levels up to 35.8%, with the next major
event at 6.7 s causing a jump in accumulated energy from 41% to 55%. The accumulated
energy tends to slowly level off near 8 s at about 60% of the total signal energy. The
minor event at 11.1 s brings the accumulated energy to 65%, followed by the event near
14 s by which time the accumulated energy has reached 75%, steadily ascending. The last
two minor events cause the accumulated energy to spike up from 89% to 93%, followed
by a slow accumulation for the next 30 s. Though lacking the same level of detail, the
secondary analysis affirms that the major pulse is associated with the fourth second and
energy levels are generally sustained up to the seventh second, rapidly falling off after 10
s. Minor events do follow, with the most distinct near 28 s. This analysis is in agreement
with the primary analysis that 50% of the signal energy is released in the first 7 s.
Based on these temporal analyses, four instantaneous spectra were extracted for
more detailed analysis in Figure 5.12. Near 4 s, marked by the arrival of the P-waves, the
primary analysis detects a very broadband spectrum holding 96% of the signal energy at
this time, indicating a large suite of contributing frequencies. Though peaking at 1.6 Hz,
the lack of frequency resolution implies that there are likely multiple distinct frequency
components at this time. The secondary analysis sheds more light on the situation. The
168
SG(a,t=13.6 s)
SG(a,t=28.0 s)
SG(a,t=4.0 s)
SG(a,t=6.5 s)
PRIMARY ANALYSIS
SG(a,t=28.0 s)
SG(a,t=13.6 s)
SG(a,t=6.5 s)
SG(a,t=4.0 s)
SECONDARY ANALYSIS
f [Hz]
f [Hz]
FIGURE 5.12. Wavelet instantaneous spectra taken at
critical time steps in the El Centro event for primary (top)
and secondary analysis
169
display has several peaks between 1 and 2 Hz, where most of the energy appears to
reside. The level of energy between 3 and 6 Hz is fairly constant and holds 30.6% of the
signal energy at this time. There is only a minor low frequency contribution centered at
0.45 Hz, with only 2.1% of the instantaneous signal energy. As listed in Table 5.5, there
are distinct components in the vicinity of 1 and 2 Hz that contribute fairly equally to the
overall instantaneous energy of the signal. In total, over 40% of the signal energy lies in
this vicinity. An additional component of high frequency energy accounts for another
12.3% of the instantaneous signal energy. At 6.5 s, the primary analysis now identifies
more distinct components, indicating that the wide array of frequencies was only present
in that first pulse. The component centered at 0.62 Hz contributes 7.1% to the energy at
this time. The second component, centered at 1.8 Hz, but essentially including
contributions from 1.6 to 2 Hz by virtue of the reduced frequency resolution, captures
51.7% of the signal energy. A third component centered at 5.2 Hz includes energy from
4.6 to 5.8 Hz, holds 33.5% of the signal energy at this time. The added detail of the
secondary analysis affirms a richness of energy from 1-3 Hz. Lumping this energy into
two modes at 1.2 and 2.2 Hz, corresponding to 23.5% and 30.7% of the instantaneous
signal energy, generally consistent with the distribution of energy noted by the primary
analysis. Again, while visible, the low frequency component centered at 0.55 Hz
contributes only 3.8% to the instantaneous signal energy. By 13.6 s, both the primary and
secondary analyses concur that there is a single mode response near 1.5 Hz associated
with over 60% of the overall instantaneous signal energy, as shown in Table 5.5. At 28 s,
the last significant pulse arrives. With its arrival, a suite of frequency components returns.
The primary analysis detects three major components. The component centered at 2.1 Hz
170
TABLE 5.5
RELATIVE CONTRIBUTIONS OF EACH COMPONENT TO INSTANTANEOUS
SPECTRA FOR EL CENTRO EARTHQUAKE PRIMARY AND SECONDARY
ANALYSES
tj
[s]
4.0
6.5
13.6
28.0
4.0
6.5
13.6
28.0
IF1
[Hz]
0.62
1.13
1.20*
Eˆ1 (t j )
[%]
7.1
11.6
12.5
IF2
[Hz]
1.60*
1.87
1.47
1.51
1.88
1.55
Eˆ 2 (t j )
IF3
[Hz]
Eˆ 3 (t j )
[%]
[%]
Primary Analysis
96.0
51.7
79.4
65.3
2.13
35.3
Secondary Analysis
16.7
2.16
13.3
2.20* 30.7
62.0
2.13
43.7
IF4
[Hz]
Eˆ 4 (t j )
5.21
33.5
4.69
56.5
4.0*
30.6
3.89*
48.6
[%]
IF5
[Hz]
Eˆ 5 (t j )
7.6
12.3
[%]
*broadband energy contribution centered or peaking at this frequency.
assimilates contributions from 1.9 to 2.4 Hz for a 35.6% share of the instantaneous signal
energy. A broader-band component near 4.7 Hz with contributions from 4.19 to 5.19 Hz
holds 56.5% of the instantaneous energy. Again, a low frequency component near 0.89
Hz is present but makes a contribution of only 4% to the energy at that time. The
resolution of the secondary analysis is able to separate more of the high frequency
components and affirms that the only significant components is actually at 2.13 Hz with
43.7% of the signal energy. A broadband characteristic, centered at approximately 4 Hz,
appears to include the contributions of two modes and nearly one half the signal energy,
as shown in Table 5.5. Though the Fourier spectrum agrees that energy is dominant
between 1-2 Hz in this signal, the intermittent presence of specific contributions to the
171
overall signal energy can only be uncovered through time-frequency analyses such as
those presented here.
Interestingly, a similar time-frequency analysis via Hilbert spectra was conducted
by Huang et al. (1998) on the same El Centro record, but with startlingly different results.
The analysis suggested that tall buildings with frequencies less than 0.5 Hz are the
primary targets of seismic energy during this event, contrary to all that was previously
observed. The authors suggest that this feature of seismic motion went previously
unnoticed as a direct consequence of the limited Fourier perspective in earthquake
engineering. In this controversial analysis, Huang et al. produced a marginal Hilbert
Spectrum with its energy focused below 5 Hz and more importantly dominating the sub0.5 Hz frequency range, in direct contrast to the Fourier spectrum. Their findings are
shown in Figure 5.13.
Based on this, Huang et al. (1998) cautioned that the “high density and low
frequency should cause major concerns for earthquake engineering, for this is exactly the
range of resonance for high-rise buildings.” However, these findings seem suspicious. As
demonstrated in Figure 5.14, a cursory inspection of the IMFs generating the Hilbert
marginal spectrum in Figure 5.13 indicates that the largest decomposed amplitudes are
associated with the first 4 IMFs, which by inspection contain predominantly higher
frequencies. (Note that the amplitude ultimately plotted in the Hilbert spectrum and its
marginal spectrum is merely the SRSS combination of the real and imaginary
components of the Hilbert transform of each IMF, i.e. the signal and its quadrature
component (This procedure is discussed in more detail later in Chapter 7). The last five
172
FIGURE 5.13. (a) Fourier spectrum and (b) low
frequency zoom; (c) Hilbert marginal spectrum and
(d) low frequency zoom for El Centro analysis
conducted by Huang et al. (1998)
IMFs appear to be associated with the frequencies of less than 1 Hz. Note that the
amplitudes of these IMFs are markedly smaller. It makes little sense then for these
components to produce the largest Hilbert spectral coefficients, as suggested by Figure
5.13.
The concerns raised by Huang et al. (1998) were based upon a marginal Hilbert
spectrum that misrepresented the nature of the seismic event. An explanation of the
marginal Hilbert spectrum is given by Huang et al. (1998):
…the frequency in either [the Hilbert Spectrum or Marginal Hilbert Spectrum]
has a totally different meaning than the Fourier spectral analysis. [In the Hilbert
Marginal Spectrum], the existence of energy at the frequency, ω, means only that,
in the whole time span of the data, there is a higher likelihood for such a wave to
have appeared locally...Consequently, the frequency in the marginal spectrum
indicates only the likelihood that an oscillation with such a frequency exists.
173
FIGURE 5.14. Ten IMF components from El Centro data provided
by Huang et al. (1998)
Viewed from this perspective, the heavy concentration of low frequency components in
the marginal Hilbert spectrum of Figure 5.13 only indicates that the low frequency
components occurred locally most persistently throughout the time history. However, this
does not imply that they carried the majority of the seismic energy, as the IMFs in Figure
5.14 affirm. As the scalogram and WIFS of the secondary analysis in Figure 5.7
demonstrate, the persistent presence of low frequency components can be rationalized,
however the detailed analysis of these components reflects that their energy contributions
are not dominant relative to other more energetic components between 1 and 2 Hz. As
discussed previously in Figure 5.9 in the context of cumulative energy in the frequency
domain, 90% of the signal energy lies below 1 Hz. Further, the instantaneous spectral
analyses discussed previously affirm that, while these components are present, their
174
contributions to each instantaneous power spectrum was on the order of a few percent,
and no components of any substance were observed below 0.4 Hz. On no account does
the wavelet analysis herein affirm Huang et al.’s findings. In fact, the wavelet marginal
spectra show good agreement overall with the Fourier representation, as expected.
It is not clear whether the calculation of the marginal Hilbert spectrum was
erroneous or if its general tendency to display “likely components” as opposed to more
“energetic components” is to blame for the counterintuitive perspectives in Figure 5.13.
Regardless, any discussions based on time-frequency analyses must be conducted with
care and verified against physical observations before unsubstantiated claims of
ignorance on the part of earthquake engineers are levied.
5.4.2 Mexico City (1985)
This magnitude 8.1 earthquake that devastated Mexico City on September 19, 1985
remains in the minds of many as one of the most tragic quakes in North American
history, as news cameras documented rescue attempts amidst the rubble of apartment
buildings and hospitals. The quake, which occurred on the western coast of the Mexican
peninsula, had tremendous impacts upon the valley of Mexico City as a result of unique
soil conditions, which amplified the low frequency components. Though the frequencies
of amplification varied from site to site within the city, the amplification was largely
between 0.2 and 0.7 Hz. The concentrations of energy in the range of 0.2-0.6 Hz, being
much higher than in previous quakes of comparable magnitude, led to heavy damage of
structures in this period range (Jennings & Sanchez-Sesma, 1989). To demonstrate the
pronounced impacts of soil conditions, a nearfield record taken near the epicenter in the
175
coastal regions of Mexico is first analyzed, recorded at Caleta de Campos, part of the
Guerrero Array. To avoid confusion, this nearfield record will be referred to as the
Guerrero MC (Mexico City) record in subsequent discussions. This coastal station had an
epicentral distance of 27.1 km, making it well suited to capturing the nearfield
characteristics of the quake. Subsequently, a free field record from the Mexico City
valley, which typifies the low frequency amplification causing much of the damage in
this quake, will also be analyzed. The nearfield Guerrero MC record is of considerable
duration, although the peak ground accelerations are not as significant. The record,
originally sampled at 200 Hz, is downsampled to 40 Hz for analysis.
The primary and secondary wavelet scalograms and WIFS for the Guerrero MC
nearfield record are presented in Figure 5.15. For reference, the mean wavelet resolutions
for each analysis are again provided in Table 5.3. The enhanced temporal resolution of
the primary analysis isolates 3 energetic bursts: one event between 10 and 15 s and the
other between 1 and 1.75 Hz, followed by strong shaking between 2 and 3 Hz
accompanied by a low frequency component near 20 s. A slight burst of energy near 21 s
is also apparent. From the scalogram and the associated WIFS, there is a collection of
energy over 30 s and spanning a robust range of frequencies. Note in the primary WIFS
that dominant ridges surface near 1.75 Hz and 0.5 Hz, the latter particularly in the
vicinity of the strong shaking at 20 s. These are flanked by intermittent contributions in
the high frequency range. The refined frequency resolution in the secondary analysis
further affirms the richness of energy and the concentrations near 2.55 Hz and 1.75 Hz
around the 20th second, as well as the precursor to the strong shaking at 12 s. Though
diminished in magnitude by virtue of the dilated temporal windows, a residual low
176
SECONDARY ANALYSIS
f [Hz]
f [Hz]
f [Hz]
f [Hz]
ag
[cm/s2]
ag
[cm/s2]
PRIMARY ANALYSIS
t [s]
t [s]
FIGURE 5.15. Guerrero MC nearfield ground motion, wavelet scalogram
and WIFS (top to bottom) for primary (left) and secondary analyses
frequency component is still apparent and further affirmed by the ridges in the secondary
WIFS. The secondary scalogram does give a better perspective on the intermittence of
high frequency components within the record, though the resolutions in the low
frequency lead to a dilation of the time windows evident by the elongated bands in the
low frequency domain. The fluctuations of these high frequency ridges become more
readily apparent in the secondary WIFS, as does the downshift of midrange frequencies
and upshift of the low frequencies approaching the strong shaking near the 20th second.
A comparison of the Fourier spectrum and wavelet marginal spectrum is provided
in Figure 5.16. From the Fourier spectrum, it is clear that there are two “tiers” of energy
177
SECONDARY ANALYSIS
SWT(fo/f), SFFT(f)
SWT(fo/f), SFFT(f)
PRIMARY ANALYSIS
f [Hz]
f [Hz]
FIGURE 5.16. Wavelet marginal spectra with Fourier power
spectrum for primary (left) and secondary analyses for Guerrero MC
nearfield record
associated with this record. The first and more energetic tier has energy from 0.25 to 5
Hz, peaking at several distinct frequencies: 0.62, 1.86, 2.83, 3.75 and 4.72 Hz. The
second tier from 5.3 to 9.6 Hz has diminished spectral amplitudes. The largest Fourier
amplitude is associated with 1.86 Hz, as shown in the plot at the left in Figure 5.17. The
wavelet marginal spectrum of the primary analysis accentuates the low frequency
components of the record, though identifying a similar band of contributing frequencies,
though with finer detail in the secondary analysis. Both analyses affirm the presence of a
low frequency component as well as the presence of energy out to 5 Hz, then rapidly
falling off by 10 Hz. The efficacy of both wavelets and Fourier transforms in capturing
the signal energy is chronicled in Table 5.4.
Figure 5.17 presents the energy accumulation rates in frequency and time for the
primary and secondary analyses. As the figure demonstrates, the primary analysis
smoothes out many of the details of the energy accumulation with frequency. The rates of
178
f [Hz]
SECONDARY ANALYSIS
t [s]
E(t)
E(f)
E(t)
E(f)
PRIMARY ANALYSIS
t [s]
f [Hz]
FIGURE 5.17. Energy accumulation in frequency domain and time domain
for primary (left) and secondary analysis of Guerrero MC nearfield record
PRIMARY ANALYSIS
dE(f)/df
dE(f)/df
SWT(fo/f)
SWT(fo/f)
SECONDARY ANALYSIS
f [Hz]
f [Hz]
FIGURE 5.18. Wavelet marginal spectrum and rate of change of energy
accumulation in frequency domain for primary (left) and secondary analysis
of Guerrero MC nearfield motion
179
change of energy accumulation in frequency and time, shown respectively in Figures 5.18
and 5.19, offer a more complete description of the energetic components of the quake.
PRIMARY ANALYSIS
dE(t)/dt
dE(t)/dt
ag
ag
SECONDARY ANALYSIS
t [s]
t [s]
FIGURE 5.19. Guerrero MC nearfield ground acceleration and rate of change
of energy accumulation in time domain for primary (left) and secondary
analysis
Due to the compromised frequency resolution in the primary analysis, the rate of
change of energy accumulation in the time domain in Figure 5.19 is able to isolate a very
energetic component at 0.47 Hz followed by a more broadband contribution peaking at
1.76 Hz. The broadness of this peak implies that a bulk of the influx of energy into the
system comes from frequencies in this vicinity. After plateauing at 4 Hz, the rate of
change of energy accumulation in the frequency domain begins to fall off steeply by 12
Hz. The secondary analysis provides some additional details on the participating
frequencies. The richness affirms that while the single most energetic component is at
0.48 Hz, a rather robust contribution from frequencies near 2 Hz is present. The peaks
associated with influxes of energy into the system can be traced back to the accumulation
180
plot in Figure 5.17 to isolate specific contributions, as shown by the dotted vertical lines.
This analysis reflects that approximately 6% of the signal energy lies at or below 0.49
Hz. Though several frequency components less than 2 Hz are shown to provide increases
to cumulative energy in the signal, a total of approximately 20% of the nearfield signal
resides less than 2 Hz. Another 20% resides at frequencies above 7.62 Hz, consistent with
the findings from simulation studies, which found that the Guerrero gap might produce
energy at a wide distribution of frequencies including high frequency motions (Jennings
et al., 1989). The primary analysis affirms that 79% of the energy associated with this
nearfield motion is associated with frequencies greater than 1.8 Hz, despite the sharp
increase in energy accumulation near 0.5 Hz. The nearfield Guerrero MC record was
found to contain only approximately 3% of its energy at frequencies under 0.5 Hz.
The intricacy of the rate of change in the time domain in Figure 5.19 highlights
one of the unique characteristics of this event. Even with the reduced resolution of the
secondary analysis, the energy influx into to the system is changing consistently between
10 to 20 s with a number of small events where the rate of energy accumulation suddenly
increases. The most noteworthy of these events is at the twentieth second. Though falling
off between 20 and 35 s, the level of activity in this event spans nearly 40 s. A number of
these isolated events can be traced back to the accumulation plots in Figure 5.17. In doing
so, it is shown that 13% of the energy arrives in the first 10.4 s. After which, the next 3 s
bring an additional 13% of the energy. The ten seconds of shaking from 14 to 24 s brings
over 46% of the signal energy, isolating the most energetic component near 20 s. The
following seven seconds bring another 14% of the signal energy before quickly falling
off beyond the thirty-first second. The accumulation in this event takes longer than the
181
other events considered in this study. The enhanced temporal resolution of the primary
analysis, if anything, affirms the flurry of activity associated with this quake. The primary
analysis generally confirms that the first 10 s is associated with only 10% of the signal
energy, while 47% of the signal energy comes in the next 10 s of shaking.
The analyses of the instantaneous spectra taken at critical moments in the
nearfield event are shown in Figure 5.20. In the primary analysis, the coarse frequency
resolution results in the lumping of energy into wider frequency bands. At 10.3 s, despite
the presence of a low frequency component near 0.5 Hz, a large concentration of energy
is associated with the high frequencies. As shown in Table 5.6, the component near 1.87
Hz, which broadly captures contributions from 1.66 to 2.08 Hz, holds 18.6% of the
instantaneous energy. The next component at 3.37 Hz representing frequency content
from 2.99 to 3.74 Hz, is associated with over one-fourth of the instantaneous energy. The
higher frequency bands from 6.42 to 8.05 Hz hold the most energy, 42.5%. The
secondary analysis, though providing added detail, generally confirms that there is a
robust distribution of energy noted in the primary analysis. Again the contributions near
0.5 Hz are just over 2%, as shown in Table 5.6. The first harmonic component of the
primary analysis is now broken into two peaks at 1.41 and 1.87 Hz, which pool between
them nearly 20% of the instantaneous signal energy. A very wide suite of energy from 3
to 6 Hz contains 33.2% of the instantaneous signal energy. A large contribution of energy
near 7 Hz is also noted. The primary analysis at the twentieth second, now associated a
more definitive share of the instantaneous energy with the mode near 0.5 Hz. By virtue of
the enhanced temporal resolution, it is apparent that the majority of energy associated
with this low frequency component is associated with the pulse at the twentieth second,
182
SG(a,t=24.7 s)
SG(a,t=33.5 s)
SG(a,t=20 s)
SG(a,t=10.3 s)
PRIMARY ANALYSIS
SG(a,t=33.5 s)
SG(a,t=24.7 s)
SG(a,t=20.0 s)
SG(a,t=10.3 s)
SECONDARY ANALYSIS
f [Hz]
f [Hz]
FIGURE 5.20. Wavelet instantaneous spectra taken at
critical time steps in the Guerrero MC nearfield event for
primary (top) and secondary analysis
183
as instantaneous spectral analyses at other time intervals fail to show such a strong
component at this frequency. Despite this contribution, as shown in Table 5.6, a majority
of the energy in this time interval remains in the high frequency ranges. The secondary
analysis affirms that the low frequency components begin to more clearly separate into
two distinct components less than 1 Hz, do retain more significance within the shaking
near the twentieth second. They are coupled with contributions of similar order (see
Table 5.6) from distinct frequencies near 1.93, 2.78 and 3.70 Hz. A very wide band of
high frequencies attributes the remaining energy in this time interval. Moving away from
the most energetic moment in the shaking, the emphasis on the low frequencies
diminishes and the energy again moves toward the higher frequencies. Energy now is
detected in significant quantities all the way to 8.62 Hz, as shown in Table 5.6, affirming
the ability of the Guerrero gap to produce such excitations. The presence of high
frequency shaking is affirmed in the secondary analysis in Table 5.6, though still
retaining the significance of the low frequency component. This may be due to the fact
that the compromised temporal resolution of this analysis leads to a smearing of energy
with time at low frequencies.
By 33.5 s, some of the low frequency content has resurfaced in the primary
analysis, though paling in comparison to the contributions at higher frequencies. The
simultaneous presence of both high and low frequency shaking is a unique characteristic
of the Mexico City event. The individual contributions are again noted in Table 5.6. The
secondary analysis affirms a decided shift toward the higher frequencies, concentrating a
majority of the instantaneous energy above 5 Hz. However, the persistent presence of the
184
TABLE 5.6
RELATIVE CONTRIBUTIONS OF EACH COMPONENT TO INSTANTANEOUS
SPECTRA FOR GUERRERO MC NEARFIELD EARTHQUAKE PRIMARY AND
SECONDARY ANALYSES
tj
[s]
IF1
[Hz]
Eˆ1 (t j )
10.3
20.0
24.7
33.5
0.49
0.49
0.56
12.1
2.56
5.54
10.3
20.0
24.7
33.5
0.49
0.49
0.49
0.56
2.2
4.91
6.05
4.50
[%]
IF2
[Hz]
1.87
1.34
1.47
1.41
0.95
1.41
Eˆ 2 (t j )
IF3
[Hz]
Eˆ 3 (t j )
[%]
[%]
Primary Analysis
16.7
1.87
18.6
5.43
11.9
Secondary Analysis
6.8
1.87
13.5
4.02
1.93
9.48
4.69
34.7
5.86
2.45
3.80
IF4
[Hz]
Eˆ 4 (t j )
Eˆ 5 (t j )
[%]
IF5
[Hz]
3.37
3.77
4.82
5.34
26.3
41.6
53.4
25.9
7.24
6.72
8.62
8.68
42.5
34.76
29.9
42.2
4.03*
2.78
5.87
5.08
33.2
17.3
11.9
21.6
7.05
3.70
9.14
8.23
36.3
13.1
24.9
46.4
[%]
*midpoint of wide band energy distribution.
low frequency contributions at 0.5 Hz should be noted, as they will play a critical role
once the seismic waves arrive in the Mexico City valley.
The collapse of a number of high-rise dwellings in the Mexico City earthquake
emphasized the significance of site conditions in enhancing seismic risk, as it was
observed that, at some locations in and near Mexico City, components in the vicinity of
0.5 Hz were dramatically amplified. Such amplification was not evident in coastal
records, as shown in the preceding analysis and in Singh et al. (1989). In fact, ground
motion at the lakebed sites in Mexico City were amplified nearly 75 times in the vicinity
of 0.5 Hz in comparison to the costal sites of equal distance from the source (Singh et al.,
1989). To demonstrate the dramatic amplification of the low frequency component, a free
field record from the Mexico City Valley is analyzed by the aforementioned primary
185
analysis technique. The mean wavelet time and frequency resolutions are 0.113 s and
0.706 Hz, respectively, and the resolution of the accompanying Fourier analysis is 0.098
Hz. Upon comparing the farfield time history in Figure 5.21, recorded in Mexico City,
with the nearfield Guerrero MC time history in Figure 5.15, the regular periodic
characteristic of the farfield record and the elongation of period are immediately evident.
An inspection of the marginal wavelet spectrum and the Fourier spectrum in Figure 5.21
demonstrates the amplification of the low frequency energy of the event that is largely
credited with the heavy damage to structures of comparable period, as noted in the postdisaster surveys of Mexico City. A strong amplification of the 0.49 Hz component is
apparent with only a slight presence at 1.46 Hz, which is comparatively negligible.
Though this 0.5 Hz component was present in the coastal Guerrero MC record analyzed
previously, the local site conditions clearly amplified this component of the shaking far
above all others. The agreement between the wavelet marginal spectrum and the Fourier
spectrum is excellent in this regard. The areas under the respective spectra also show
good agreement with the signal variance. The signal had a standard deviation of 29.13
cm/s2 in comparison to the 29.61 cm/s2 estimated from the Fourier spectrum and the
29.16 cm/s2 from the wavelet spectrum. The enhanced low frequency resolution of the
wavelet analysis is particularly beneficial in capturing the energy of this record.
The scalogram also in Figure 5.21 affirms the concentration of energy at this
frequency and identifies short duration pulses preceding the major pulse. The ridge
extraction of the WIFS detects a subtle shifting of frequency. In comparison to the
nearfield analysis in Figure 5.15, it becomes apparent that the amplification of the 0.5 Hz
component dwarfs any residual contributions at the many other frequencies present in the
186
ag
f [Hz]
f [Hz]
SWT(fo/f), SFFT(f)
f [Hz]
t [s]
FIGURE 5.21. Comparison of Fourier spectrum and wavelet marginal
spectrum (left) for Mexico City farfield ground motion with wavelet
scalogram and WIFS (right, top to bottom)
quake. The energy accumulations in time and frequency and the rate of change in time
are provided in Figure 5.22. The rate of change in the frequency domain (not shown),
peaks at 0.47 Hz, as can be inferred from the accumulation plot in Figure 5.22 that has a
steep descent in this vicinity. From this plot, it is demonstrated that 80% of the energy at
this site comes from frequencies between 0.37 and 0.67 Hz. An additional 10% of the
energy is associated with frequencies less than 0.37 Hz. The rate of change of energy
accumulation in the time domain is actually more insightful, isolating the three pulses in
the quake at 42.5s, 50.4 and 58.5 s, which happens to be most energetic. Relaying this
information to the energy accumulation, it is demonstrated that one-quarter of the signal
energy arrives with this main pulse. To determine the frequency content of the quake at
187
ag
dE(t)/dt
E(t)
E(f)
f [Hz]
t [s]
t [s]
FIGURE 5.22. Energy accumulation in frequency and time (left),
accompanied by Mexico City nearfield ground acceleration and rate of change
of energy accumulation in time domain
these various time intervals, several instantaneous spectra are assembled in Figure 5.23.
At each of the time steps associated with peaks in the rate of change of energy
accumulation in Figure 5.22, the single component at 0.49 Hz dominates, with
approximately 98% of the instantaneous signal energy at that particular time interval.
Only after the shaking has subsided at 79.4 s does another frequency component surface,
as approximately 20% of the energy surfaces at 1.47 Hz, the component just barely
visible in the Fourier spectrum presented previously. This analysis affirms that shaking
associated with the main three pulses are strongly focused at 0.5 Hz, while it is only in
the residual shaking afterward that the additional component at 1.47 Hz emerges. The
complexity of the instantaneous power spectrum in this latter stage of shaking reflects
that under the diminished shaking, the amplitude of the 0.5 Hz begins to approach the
energy levels associated with other harmonics.
188
SG(a,t=24.7 s)
SG(a,t=24.7 s)
SG(a,t=24.7 s)
SG(a,t=24.7 s)
f [Hz]
f [Hz]
FIGURE 5.23. Wavelet instantaneous spectra
taken at critical time steps in the Mexico City
farfield record
5.4.3 Loma Prieta (1989)
An earthquake measuring 7.1 Ms occurred 18 km beneath the Santa Cruz Mountains in
California on October 17, 1989, at 17:04 local time. The California Strong Motion
Instrumentation Program (CSMIP) station closest to the epicenter of this so-called Loma
Prieta event was the Corralitos station, located approximately 10 km away and very close
to the San Andreas fault near the center of the aftershock zone. At this site, 0.64 g of
motion was detected. At that time, this was the first nearfield record of a magnitude 7
event (Benuska, 1990). The ground motion record, with peak acceleration (PGA) of
0.6297 g, was fortunately relatively short in duration.
Again the two standard wavelet analyses used previously are presented in tandem
for this event. The mean wavelet resolutions are given in Table 5.3. The signal was
189
downsampled from its original sampling rate of 50 Hz to 25 Hz to ease computation.
Figure 5.24 shows the wavelet scalograms and the accompanying wavelet instantaneous
frequency spectra for both analyses. Both the scalograms and WIFS affirm that the
energy is primarily released within the first 10 s of the event. The primary analysis
provides better temporal information and affirms that the energy lies between 2-10 s and
concentrates at approximately 2.7-3 s, with the arrival of the largest pulse. The primary
analysis detects a broad range of frequency contributions near 1 Hz and spanning all the
way to nearly 4 Hz. Some secondary energy arrives near 5 and 7 s but pales in
comparison to this initial pulse. The ridges in the primary analysis detect some slight high
and low frequency contributions, which are minor in contrast to the persistent frequency
contributions in the vicinity of 1.15 Hz, 1.5 Hz and 2.25 Hz. This latter component
demonstrates a shift toward 3 Hz and then softens in frequency. Note that the most
energetic burst at 2.8 s is tied to this higher frequency component.
The secondary analysis affirms that the primary burst of energy is concentrated
near 2.75 Hz, though there are now more distinct bands of frequency evident, by virtue of
the enhanced resolution, at approximately 1.25 Hz, 1.75 Hz, 2.25 Hz and 3 Hz.
Interestingly these latter two components begin to approach one another, the 3 Hz
softening toward the stiffening 2.75 Hz component. This stiffening and softening was
crudely captured in the primary analysis by the increase and drop in the higher frequency
component peaking near 3 Hz. The energy associated with higher frequencies between 3
and 4 Hz also is readily more apparent.
190
SECONDARY ANALYSIS
f [Hz]
f [Hz]
f [Hz]
f [Hz]
ag
[cm/s2]
ag
[cm/s2]
PRIMARY ANALYSIS
t [s]
t [s]
FIGURE 5.24. Loma Prieta ground motion, wavelet scalogram and WIFS (top to
bottom) for primary (left) and secondary analyses
A comparison of the Fourier power spectrum and the marginal wavelet spectrum
for each analysis is provided in Figure 5.25. The coarser primary analysis lumps the
signal energy into two dominant bands of energy near 1.5 Hz and 3 Hz. However,
revisiting Table 5.3, the frequency resolution of the secondary wavelet analysis is more
akin to the resolution of the Fourier spectrum. As such, this secondary analysis detects
many of the finer spectral details of the Fourier analysis. The Fourier analysis identifies a
dominant contribution near 1.4 Hz coupled with an energetic component near 2.8 Hz.
These two dominant modes are similar to those specified by the primary wavelet
marginal spectrum. The differences in amplitudes are merely the function of the finer
191
SECONDARY ANALYSIS
SWT(fo/f), SFFT(f)
SWT(fo/f), SFFT(f)
PRIMARY ANALYSIS
f [Hz]
f [Hz]
FIGURE 5.25. Wavelet marginal spectra with Fourier power
spectrum for primary (left) and secondary analyses for Loma Prieta
quake
bandwidth in the secondary wavelet analysis directly accounted for in the scaling by a.
Still the energy is completely conserved in the wavelet representation and even more so
than the Fourier analysis, as evidenced by Table 5.4 when comparing to the signal
standard deviation σ. Note that at the low frequencies, the superior resolution of the
wavelet provides some added detail. In general, both the wavelet and Fourier spectra
confirm that there is only trace energy below 1 Hz and above 4 Hz.
The energy accumulation plots in Figure 5.26 account for the multi-resolution
characteristics of wavelets and provide a more objective tool for investigating the
dominant energy components in the signal in comparison to the magnitude of wavelet
scalograms. The dotted lines on the energy accumulation with frequency are associated
with the most significant spectral peaks identified in the primary and secondary analyses.
In the primary analysis, the energy accumulated up to 2.8 Hz is 46%, and by 1.5
Hz, 79% of the signal energy has been expelled. The four major peaks identified by the
192
SECONDARY ANALYSIS
f [Hz]
E(t)
E(f)
E(t)
E(f)
PRIMARY ANALYSIS
f [Hz]
t [s]
t [s]
FIGURE 5.26. Energy accumulation in frequency domain and time domain
for primary (left) and secondary analysis of Loma Prieta
secondary analysis verify that at 3.8 Hz, only 16% of the signal energy has been realized.
By 2.8 Hz, this has jumped to 42% followed by a steep vertical ascent to 50% of the
signal energy. This would indicate that nearly 35% of the total signal energy is associated
with this component of the spectrum. 70% of the signal energy lies above 1.8 Hz with
81% of the energy above 1.4 Hz again followed by a very steep ascent to nearly 90% of
the signal energy. Thus 20% of the signal energy is associated with this lower frequency
component. The vertical ascents at 2.8 and 1.4 Hz are indicative of very rapid changes in
signal energy. The frequencies at which these occur can be considered the most energetic
frequency components in the signal. This can be more readily visualized by examining
the rates of change of energy accumulation, shown for frequency in Figure 5.27. For the
primary analysis, it becomes evident that the most energetic frequency component is
associated with 1.5 Hz, which corresponds to the dominant frequency component in the
Fourier spectrum. An additional energetic component at 2.8 Hz is also affirmed, along
with a minor low frequency presence at 0.5 Hz. The accumulation curves in Figure 5.28
193
PRIMARY ANALYSIS
dE(f)/df
dE(f)/df
SWT(fo/f)
SWT(fo/f)
SECONDARY ANALYSIS
f [Hz]
f [Hz]
FIGURE 5.27. Wavelet marginal spectrum and rate of change of energy
accumulation in frequency domain for primary (left) and secondary analysis
of Loma Prieta
affirm that 90% of the signal energy is above approximately 1.25 Hz. In the secondary
analysis, the findings are quite similar though more details are resolved. Still the energy
is harnessed at 1.8 Hz and additionally near 3 Hz. The highest rates of change in energy
accumulation are appropriately associated with the marginal spectral peaks and the
analysis provides no evidence of significant high or low frequency contributions.
The rate of change of energy accumulations in the time domain is provided in
Figure 5.28. The superior time resolution of the primary analysis is evident. Not only is
the arrival of the major pulse at 2.8 s clearly identified, but also the sustaining of energy
between 3 and 8 s, with the exception of occasional spikes at 4 and 5.5 Hz. The
termination of the major energetic components occurs near 8 s. After this point, the
energy drops off steeply and has hardly any contributions beyond 9 s. Returning to the
energy accumulation in Figure 5.26, the dotted line indicates the arrival of the major
194
PRIMARY ANALYSIS
dE(t)/dt
dE(t)/dt
ag
ag
SECONDARY ANALYSIS
t [s]
t [s]
FIGURE 5.28. Loma Prieta ground acceleration and rate of change of energy
accumulation in time domain for primary (left) and secondary analysis
pulse where signal energy jumps from 25% to 30% and sustains its intensity over the first
5 s, as over half the signal energy arrives. By nearly 7.8 s, 90% of the energy has passed,
accounting for the drop off in dE(t)/dt beyond 8 s. The secondary analysis is incapable of
providing such detail. It captures the major event at 2.8 s and the cumulative energy plot
affirms a jump from 24% signal energy to 33%, followed by a rather gradual decline in
energy terminating at 10 s. This loss of resolution is the consequence of the trade-offs
between time and frequency resolution. Still the analysis sufficiently gauges the
localization of energy, with the times and frequencies associated with the maximum rates
of change in energy accumulation representing the most significant contributors to the
event.
Four instantaneous spectra were extracted at key moments in the progression of
the earthquake, as shown in Figure 5.29. At 2.8 s, the arrival of the major pulse of the
quake, the primary analysis detects a bi-modal response peaking at 1.7 and 3.4 Hz. These
two components respectively contribute 16.7 and 78.7 % of the energy at this time. The
195
SG(a,t=5.5 s)
SG(a,t=8.0 s)
SG(a,t=4.0 s)
SG(a,t=2.8 s)
PRIMARY ANALYSIS
SG(a,t=8.0 s)
SG(a,t=14.0 s)
SG(a,t=5.5 s)
SG(a,t=2.8 s)
SECONDARY ANALYSIS
f [Hz]
f [Hz]
FIGURE 5.29. Wavelet instantaneous spectra taken at critical time
steps in the Loma Prieta event for primary (top) and secondary
analysis
196
enhanced resolution of the secondary analysis at this same time frame identifies two
groupings of frequency contributions, each manifesting multiple peaks. The major
grouping near 1.6 Hz holds 21% of the instantaneous signal energy and the grouping near
3 Hz holds 71% of the instantaneous energy at this time. These statistics are chronicled in
Table 5.7. Both analyses confirm that this first and most energetic pulse contains
dominant energy in the vicinity of 3 Hz. The next instantaneous spectrum at
approximately 4 s, associated with the minor event, shows a slight low frequency
component but two major peaks at 1.27 and 2.91 Hz, a down shift in frequency from the
earlier pulse. Still the energy contributions are primarily associated with the higher
frequency component (82.4%), with only 11.1% associated with the 1.27 Hz peak. By 5.5
s, the primary analysis continues to display two dominant modes with some minor low
frequency content. The two peaks shift to the higher frequencies of 1.43 and 3.56 Hz with
the arrival of this third event, also denoted in Figure 5.29. More energy is present in the
lower frequency peak (31.6%) than previously observed, though the 3.56 Hz component
still garners the most energy (63.6%). For the secondary analysis, this is better resolved
to identify three major clusters of frequency contributions 1.5 Hz, 2.8 Hz and 3.9 Hz with
respective instantaneous energy of 37%, 16% and 27%. While the primary and secondary
analysis are in agreement with the energy contained in the 1.5 Hz vicinity, the increase in
resolution identifies in greater detail that the energy in the higher frequencies is
predominantly near 3.9 Hz.
By the 8th second, energy is known to drop off dramatically. The primary analysis
identifies two components at 1.51 and 3.89 Hz, again shifting to higher frequencies, and
energy now associated more so with the lower frequency component, 65.3% vs. 32.2%.
197
TABLE 5.7
RELATIVE CONTRIBUTIONS OF EACH COMPONENT TO INSTANTANEOUS
SPECTRA FOR LOMA PRIETA EARTHQUAKE PRIMARY AND SECONDARY
ANALYSES
tj
[s]
IF1
[Hz]
Eˆ1 (t j )
[%]
IF2
[Hz]
2.8
4.0
5.5
8.0
1.70
1.27
1.43
1.51
2.8
5.5
8.0
14.0
1.59
1.50
1.43
1.52
0.70
15.6
Eˆ 2 (t j )
IF3
[Hz]
Eˆ 3 (t j )
[%]
[%]
Primary Analysis
16.7
11.1
31.6
65.3
Secondary Analysis
71.0
37.0
60.0
23.7
2.50
14.9
IF4
[Hz]
Eˆ 4 (t j )
3.40
2.91
3.56
3.89
78.7
82.4
63.6
32.2
3.0
2.8
3.32
IF5
[Hz]
Eˆ 5 (t j )
21.0
16.0
3.9
27.0
20.7
3.97
15.6
[%]
[%]
In the secondary analysis, the diminishing high frequency components are affirmed, as
the instantaneous spectrum falls into a single modal response at 1.43 Hz, harnessing 60%
of the instantaneous energy. The secondary analysis is extended to 14 s, where the energy
release has mostly subsided. The response here still concentrates at 1.52 Hz (with 23.7%
of the instantaneous energy), accompanied by a 0.70 Hz component (15.6%), a 2.50 Hz
component (14.9%), a 3.32 Hz component (20.7%) and a 3.97 Hz component (15.6%). In
terms of energy distribution, the energy of the quake following strong shaking is
relatively broadband. A primary analysis (not shown) at this same time affirmed shared
energy over a range of frequencies, with 28% localized near 1.5 Hz and 42% between
2.32 and 3.67 Hz, in general agreement with the secondary analysis.
When interpreting the primary analysis results, it is important to recall that a
wider range of frequencies is contributing to the energy near 3 Hz, from 2.32 to 3.67 Hz,
198
giving it a consistently dominant share of the instantaneous energy. The secondary
analysis identifies many contributors in this range whose individual share of the
instantaneous energy generally sums to the energy associated with the higher frequency
mode of the primary analysis. However, the rates of change of energy accumulation
affirm that the frequencies near 1.8 Hz are responsible for the most sudden influx of
energy. This is the single most dominant component of the quake, as also affirmed by
Fourier, while the most dominant range of frequencies contributing to the shaking is near
3 Hz, corresponding to the spectral contributor with the greatest “bandwidth,” a
characteristic that also can be affirmed from the Fourier spectrum in Figure 5.25.
5.4.4 Northridge (1994)
The 6.7 magnitude Northridge earthquake on January 17, 1994 represented California’s
most devastating earthquake in recent history. Striking at 04:30 local time, collapses of
residential dwellings caught many victims in their sleep, prompting a wave of seismic
research to uncover the sources of numerous failures and deficiencies in constructed
facilities. The particular record analyzed herein was taken 2.31 km from the epicenter,
representing nearfield, strong shaking. The signal was sampled originally at 50 Hz and
then downsampled to 25 Hz for the following wavelet analysis, whose resolutions are
provided in Table 5.3.
As demonstrated by the primary analysis scalogram in Figure 5.30, with time, the
energy associated with each pulse tends toward higher frequencies in a general hardening
trend, observed near 4 s and 8 s. The first event has a distinctly higher frequency content,
centered around 2 Hz, while the latter event shifts to 1.3 Hz. The majority of the signal
199
SECONDARY ANALYSIS
f [Hz]
f [Hz]
f [Hz]
f [Hz]
ag
[cm/s2]
ag
[cm/s2]
PRIMARY ANALYSIS
t [s]
t [s]
FIGURE 5.30. Northridge ground motion, wavelet scalogram and WIFS (top to
bottom) for primary (left) and secondary analyses
energy is released within the first 10 s of the event. Though detecting a slight high
frequency contribution, the WIFS affirms the dominance of energy between 1 and 2 Hz.
The two signatures at 4 and 8 s share an ascending characteristic with neighboring
frequency bands. Persistent components at 0.5 and 0.8 Hz are also noteworthy.
In the secondary analysis, also shown in Figure 5.30, the temporal bleeding noted
previously is again present. The enhancement of frequency resolution isolates a few
distinct frequency bands in the scalogram. The first event tends to focus primarily in the
2.3 Hz range. The latter event has more of a concentration near 2.21 Hz, flanked by a
0.76 Hz component. The abundance of low frequency ridges in the secondary WIFS may
200
be a direct consequence of an over resolution in the low frequency range. It does however
detect two mirroring ridges near 2 and 3 Hz.
The Fourier spectrum in Figure 5.31 indicates that a wide band of energy from
0.46 to 3.01 Hz provides the driving energy in the earthquake. The largest Fourier
amplitudes are near 2.3 Hz, but there also is a strong component near 0.8 Hz, as
mentioned previously in the discussion of the scalograms. Beyond 3 Hz, the energy
rapidly falls off. The wavelet marginal spectrum in the primary analysis captures largely
the same trends, though the relative magnitudes emphasize the lower frequencies, as
expected. The wider bandwidth of the components greater than 1 Hz implies that their
amplitudes must be reduced in order to conserve energy. As noted in Table 5.4, energy is
better captured by the primary wavelet analysis. The added detail in the secondary
marginal wavelet spectrum reveals three contributing bands of energy near 0.6 Hz, 1.7 Hz
and 2.4 Hz. These are consistent with the bands observed at different times in the wavelet
scalogram and will be discussed in more detail in the instantaneous spectral analysis. The
enhanced resolution in the secondary analysis, permits a better estimation of the overall
energy, as shown in Table 5.4.
The energy accumulation plots in Figure 5.32 will now be discussed in parallel
with the energy accumulation rates of change in the frequency and time domains, shown
respectively in Figures 5.33 and 5.34. As noted previously, diminished resolutions in the
time and frequency domains tend to smooth out the details of the energy accumulation
plots, as seen from the primary and secondary analyses in Figure 5.32. In the frequency
domain, as shown in Figure 5.33, the primary analysis, lacking sufficient detail, identifies
201
PRIMARY ANALYSIS
SWT(fo/f), SFFT(f)
SWT(fo/f), SFFT(f)
SECONDARY ANALYSIS
f [Hz]
f [Hz]
FIGURE 5.31. Wavelet marginal spectra with Fourier power spectrum
for primary (left) and secondary analyses for Northridge quake
f [Hz]
SECONDARY ANALYSIS
E(t)
E(f)
E(t)
E(f)
PRIMARY ANALYSIS
f [Hz]
t [s]
t [s]
FIGURE 5.32. Energy accumulation in frequency domain and time domain for
primary (left) and secondary analysis of Northridge
202
SECONDARY ANALYSIS
dE(f)/df
dE(f)/df
SWT(fo/f)
SWT(fo/f)
PRIMARY ANALYSIS
f [Hz]
f [Hz]
FIGURE 5.33. Wavelet marginal spectrum and rate of change of energy
accumulation in frequency domain for primary (left) and secondary analysis of
Northridge
SECONDARY ANALYSIS
dE(t)/dt
dE(t)/dt
ag
ag
PRIMARY ANALYSIS
t [s]
t [s]
FIGURE 5.34. Northridge ground motion and rate of change of energy
accumulation in time domain for primary (left) and secondary analysis
203
a large band of frequencies associated with the release of energy during the quake. This
would indicate that a suite of frequencies from 0.4 to 2.7 Hz contributes in a comparable
manner to the shaking. The frequency associated with the greatest change in energy
accumulation is 0.76 Hz, closely followed by the component at 2.25 Hz. The bandwidths
of contributing frequencies as well as the two aforementioned peak frequencies are
denoted in Figure 5.32 by dotted lines. The energy accumulation plot in the time domain
reflects that 27.3% of the energy lies above 2.76 Hz. 14% of the energy is between 2.3
and 2.7 Hz, while 40% of the energy is between 0.78 and 2.3 Hz, the broad range in
which most of the scalogram energy was focused. Approximately 10% of the signal
energy lies below 0.78 Hz.
The secondary analysis in Figure 5.33 provides some additional insights. Two
distinct contributions to the rate of change of energy accumulation are observed at 0.48
and 0.79 Hz, followed by a constant influx of energy between 1.3 and 1.8 Hz. This is
trailed by another influx of energy at 2.4 Hz. Again relating these particular frequency
components to the energy accumulations in Figure 5.32 reveals that 38.4% of the signal
energy is associated with frequencies higher than 2.4 Hz, while 13% are associated with
frequencies from 1.9 to 2.4 Hz. The bands from 1.3 to 1.8 Hz and 0.8 to 1.3 Hz both are
associated with approximately 12% of the signal energy. Only 6% of the energy is
associated with 0.5-0.8 Hz. These analyses indicate that the dominant energy of the signal
lies between 0.8-2.4 Hz, in agreement with the Fourier spectrum. However, the most
energetic frequency contribution comes from 0.78 Hz. This frequency brings the single
greatest influx of energy to the signal – a characteristic not evident from the Fourier
spectrum.
204
From the time rate of change in Figure 5.34, the primary analysis denotes a
constant energy contribution from 3.3 to 9.6 s, during which there are four distinct events
at 4.24, 5.72, 7.64 and 8.92 s. These are the peak events creating the energy ascent
demonstrated previously in the scalogram. These instances can be correlated back to the
energy accumulation plots in Figure 5.32, as denoted by the dotted lines. This analysis
indicates that 15% of the energy arrives in the first 4.14 s, at which point the cumulative
energy jumps another 5%, marking the time with the highest energy influx. In the next
two seconds, another 16% of energy is released. The following two seconds provide
another 20% jump in energy. By 7.6 s, there is a spike in energy of 3% and then a 14%
rise in energy up to 8.9 s. By this point, 75% of the quake’s energy has been released,
affirming the concentration of energy in the scalogram under 10 s. The secondary
analysis smoothes out many of these details, as shown in Figure 5.34. Needless to say,
the energy is still shown to lie between 3 and 10 s, peaking at 4.86 s, with a lesser peak at
8.95 s. A similar inspection of the accumulation plot in Figure 5.36 demonstrates that
only 6.5% of energy lies in the first 3 s of shaking. The majority of the energy (45%) is
released between 4.8 and 9 s, consistent with the primary analysis.
The primary instantaneous spectral analysis in Figure 5.35 provides more insights
into the spectral distribution of energy during different critical events in the Northridge
quake. The first spectrum, associated with 4.2 s, identifies one major energy component
at 2.26 Hz, accompanied by some lower frequency content. As shown in Table 5.8,
88.9% of the instantaneous signal energy is associated with this single component. By
5.72 s, this low frequency component becomes more energetic, holding 18.37% of the
instantaneous energy. At the same time, the dominant mode has stiffened and holds
205
SG(a,t=8.9 s)
SG(a,t=7.6 s)
SG(a,t=5.7 s)
SG(a,t=4.2 s)
PRIMARY ANALYSIS
SG(a,t=8.9 s)
SG(a,t=7.6 s)
SG(a,t=5.7 s)
SG(a,t=4.2 s)
SECONDARY ANALYSIS
f [Hz]
f [Hz]
FIGURE 5.35. Wavelet instantaneous spectra taken at
critical time steps in the Northridge event for primary (top)
and secondary analysis
206
72.5% of the instantaneous energy. By 7.6 s, the response has focused toward a single
mode at 1.8 Hz holding most of the energy, as shown in Table 5.8. Concurrently, the
lower frequency mode has also shifted toward lower frequencies and holds 12.6% of the
instantaneous signal energy. This tendency was reflected in the scalogram through a shift
of energy toward lower frequencies in the two consecutive major events. By 8.9 s, the
energy has been halved, though a low frequency mode at 0.62 Hz remains, with its
energy content somewhat diminished. The combination of contributions at 1.8 and 2.3 Hz
now hold 64.3% of the signal energy, with the addition of a high frequency component at
5 Hz, which represents a range of energy from 4.5 to 5.5 Hz. These analyses again
demonstrate the importance of understanding that the energy under instantaneous spectra,
and not the spectral amplitude, should be consulted to determine the dominant frequency
within a signal.
Marginal spectra demonstrated larger amplitudes at the lower
frequencies, but the detailed analysis of energy distribution affirms that the dominant
component in the quake was actually near 2.26 Hz.
The secondary analysis, also shown in Figure 5.35, provides potential over
resolution of the low frequency components, but still discerns a dominant mode at 2.46
Hz at 4.2 Hz and is capable of separating out an additional mode at 3.11 Hz, whose
respective energy contributions are given in Table 5.8. These two higher frequency
modes then collapse into one peaking at 2.52 Hz and again comprising the primary
energy source for the quake, accompanied by a broad band of energy in the low
frequency range. By 7.6 s, these low frequency components become more distinct but
still reiterating the strength of contributions in the vicinity of 2 Hz. At 8.9 s, the same
207
TABLE 5.8
RELATIVE CONTRIBUTIONS OF EACH COMPONENT TO
INSTANTANEOUS SPECTRA FOR NORTHRIDGE EARTHQUAKE
PRIMARY AND SECONDARY ANALYSES
tj
[s]
IF1
[Hz]
Eˆ1 (t j )
4.2
5.7
7.6
8.9
0.82
0.75
0.62
18.4
12.6
6.3
4.2
5.7
7.6
8.9
0.75*
0.75
0.75
0.75
28.9
34.0
9.2
7.7
[%]
IF2
[Hz]
Eˆ 2 (t j )
IF3
[Hz]
[%]
Primary Analysis
2.26
2.72
1.80
84.0
1.80
64.3
2.26
Secondary Analysis
2.46
2.52
1.74
41.3
2.39
1.67
30.0
2.39
Eˆ 3 (t j )
IF4
[Hz]
Eˆ 4 (t j )
**
5.0
23.0
43.6
47.0
28.3
26.2
3.11
16.0
4.88
17.3
[%]
[%]
88.9
72.5
*Broadband energy peaking at this frequency.
**Shares energy in one broad peak with previous frequency component.
contributors remain, joined however by an additional component near 5 Hz, whose
instantaneous spectral energy contributions are listed in Table 5.8.
5.4.5 Kobe (1995)
One year after Northridge, on January 17, 1995 at 05:46 local time, a magnitude 7.2
earthquake hit Kobe, Japan, an area viewed previously to have relatively low seismic
risk. The Nojima fault triggered the earthquake and subsequent rupture of several smaller
faults in the Kobe area. The loss of 5600 lives and destruction of 100,000 buildings
affirmed the devastation of this event. The Takatori N-S Kobe ground motion, measured
approximately 10 km from the epicenter, is analyzed herein. The signal, sampled
originally at 100 Hz, is downsampled to 25 Hz and analyzed by the aforementioned
208
wavelets to produce the scalograms and WIFS in Figure 5.36. Note the strong period of
shaking in the time history in Figure 5.36 for the first 15 s, followed by a regular,
periodic motion.
Both the primary and secondary analyses reiterate the same general finding: the
Kobe event focused its dominant energy below 1 Hz. The primary analysis further
affirms the arrival of a large low frequency pulse near 5 s and even lower frequency
shaking subsequently. The scalogram is generally devoid of higher frequency content.
The last meaningful shaking subsides by 15 s. The low frequency energy is flanked at
three distinct locations by energy in the 2 Hz vicinity, in the primary event and in
subsequent pulses near 10 and 15 s. The WIFS merely affirms the predominance of
shaking below 1 Hz. In the secondary analysis, two predominant ridges are more clearly
defined near 0.78 and 1.3 Hz. At the major pulse of the quake, some additional frequency
components between these two also surface. This plethora of ridges may be the
consequence of over resolution in the low frequency range. After 5 s, the components
near 2 Hz split and move toward higher and lower frequencies.
Analysis of the traditional Fourier spectrum in Figure 5.37 affirms that the energy
is focused in a narrowband between 0.4 and 1 Hz., peaking at a value of 0.9 Hz. The
presence of minor yet constant energy between 1 and 3 Hz is also noted, though this
energy falls off rapidly beyond 3 Hz. Both the wavelet marginal spectra and Fourier
spectra in the primary and secondary analyses capture the frequency content of the signal.
The amplification of amplitudes and the added detail in the higher frequencies in the
secondary analysis is a consequence of the enhanced frequency resolution in the
209
ag
[cm/s2]
SECONDARY ANALYSIS
f [Hz]
f [Hz]
f [Hz]
f [Hz]
ag
[cm/s2]
PRIMARY ANALYSIS
t [s]
t [s]
FIGURE 5.36. Kobe ground motion, wavelet scalogram and WIFS (top to
bottom) for primary (left) and secondary analyses
PRIMARY ANALYSIS
SWT(fo/f), SFFT(f)
SWT(fo/f), SFFT(f)
SECONDARY ANALYSIS
f [Hz]
f [Hz]
FIGURE 5.37. Wavelet marginal spectra with Fourier power spectrum
for primary (left) and secondary analyses for Kobe quake
210
secondary analysis. The conservation of energy of all the spectral representations is again
demonstrated in Table 5.4.
The energy accumulation plots in Figure 5.38, when viewed in conjunction with
the energy accumulation rates of change in the frequency and time domains, shown
respectively in Figures 5.39 and 5.40, provide a comprehensive perspective on the time
and frequency characteristics and their energy contributions to the signal. In Figure 5.39,
the rate of change of energy accumulation in the frequency domain from the primary
analysis demonstrates a concentration of energy between 0.45 and 1.0 Hz, peaking at
0.76 Hz. Though some minor energy is infused between 1.5 and 3 Hz, the energy
associated with this quake is concentrated in a relatively narrow band. The secondary
analysis affirms this and provides more detail, identifying two events in that narrow band
and then sharply dropping off at 1 Hz. The two peaks accentuated by the finer frequency
resolution of the secondary analysis are at 0.54 Hz and 0.84 Hz. The frequencies are
demarcated by dotted lines in the accumulation plots in Figure 5.38. From the
accumulation plots in the primary analysis, the rapid ascent in energy is apparent. At 1
Hz, the accumulated energy rises from 38% to 41%. Further analysis shows that only 3%
of the signal energy resides under 0.4 Hz. 20% of the signal energy lies between 0.4 and
0.76 Hz, and the frequency range of 0.76 Hz to 1 Hz holds 25% of the energy. As
demonstrated by Figure 5.38, more than half the energy in the primary analysis is less
than 1 Hz. The largest single jump in frequency occurs near 0.76 Hz, where the
accumulated energy ascends from 66 to 74%. The secondary analysis confirms that 36%
of the energy resides at frequencies greater than 1 Hz, in general agreement with the
primary analysis. Further about 20% of the energy lies between 0.84 Hz and 1 Hz and
211
f [Hz]
SECONDARY ANALYSIS
E(t)
E(f)
E(t)
E(f)
PRIMARY ANALYSIS
t [s]
f [Hz]
t [s]
SECONDARY ANALYSIS
dE(f)/df
SWT(fo/f)
PRIMARY ANALYSIS
dE(f)/df
SWT(fo/f)
FIGURE 5.38. Energy accumulation in frequency domain and time domain
for primary (left) and secondary analysis of Kobe
f [Hz]
f [Hz]
FIGURE 5.39. Wavelet marginal spectrum and rate of change of energy accumulation
in frequency domain for primary (left) and secondary analysis of Kobe
212
SECONDARY ANALYSIS
dE(t)/dt
dE(t)/dt
ag
ag
PRIMARY ANALYSIS
t [s]
t [s]
FIGURE 5.40. Kobe ground acceleration and rate of change of energy
accumulation in time domain for primary (left) and secondary analysis
between 0.54-0.84 Hz, respectively. Ten percent of the signal energy is found at
frequencies less than 0.54 Hz. The accumulated energy increases most drastically at 0.84
Hz, where it rises from 57.4% to 63.0%. The next major rise in energy is at 0.54 Hz,
where the accumulated energy jumps from 86.4% to 92%.
Referring now to the rate of change in accumulated energy in the time domain, in
Figure 5.40, the primary analysis affirms that a first event arrives near 3 s and is followed
by the major event at 4.58 s. The energy rates fall of quickly until the arrival of a smaller
event at 13.2 s, after which the infusion of energy falls quickly to zero within 20 s.
Noting these critical time instants, an analysis of the accumulated energy in Figure 5.38
reveals that the event at 3.16 s was associated with only 3% of the overall energy.
Subsequently, in the next two seconds the accumulated energy quickly rises another 10%.
With the arrival of the event at 13.2 s, over 90% of the quake’s energy has been recorded.
Many of the details of the arrival of individual events are obscured in the secondary
213
analysis, due to the temporal smoothing. Still, the major pulse is identified at 4.06 s, after
which the energy falls off, in tiers after 5 and 10 s.
A more careful analysis of the contributions by the various frequency components
at particularly interesting sequences of the quake is provided in Figure 5.41 through the
wavelet instantaneous power spectra. At 3.0 s, the primary analysis identifies 3 major
components at 0.52, 0.84 and 2.60, with the former two being very closely coupled. Their
respective energy contributions are tabulated in Table 5.9. They indicate that the low
frequency component at 0.84 Hz is very strong. The component at 2.6 Hz, which
represents contributions from 2.3 to 2.9 Hz, carries a significant portion of the
instantaneous signal energy.
The secondary analysis, with a more refined resolution, gives a more precise
depiction of the higher modes, identifying an energetic band between 1.5 and 2 Hz. A
distinct mode at 2.67 Hz surfaces, carrying 22.2% of the instantaneous spectral energy.
The lower frequency component at 0.84 Hz actually dominates in this analysis, carrying
27.6% of the instantaneous spectral energy. At 4.5 s, the energy has shifted in the primary
analysis to one lower frequency mode at 0.8 Hz accompanied by a higher mode at 1.8 Hz.
41.2% of the instantaneous spectral energy resides in the narrow frequency band
represented by the 0.8 Hz component. In the secondary analysis, the energy associated
with 2.6 Hz has diminished somewhat, but the remaining characteristics of the spectra are
largely unchanged from what was observed at 3 s. The coupled low frequency
components in the primary analysis are still present at 7.4 s, with approximately 75% of
the signal energy in this band of frequencies between 0.5 and 1 Hz. The presence of the
214
SG(a,t=7.2 s)
SG(a,t=13.0 s)
SG(a,t=4.5 s)
SG(a,t=3.0 s)
PRIMARY ANALYSIS
SG(a,t=7.2 s)
SG(a,t=13.0 s)
SG(a,t=4.5 s)
SG(a,t=3.0 s)
SECONDARY ANALYSIS
f [Hz]
f [Hz]
FIGURE 5.41. Wavelet instantaneous spectra taken at critical
time steps in the Kobe event for primary (top) and secondary
analysis
215
third mode at 2.38 Hz now becomes more visible, making a 12.5% contribution to the
instantaneous spectral energy. The secondary analysis results shown in Table 5.9 confirm
these findings and noting that the energy at this point is shifting to lower frequencies than
the previous times analyzed. By the thirteenth second, the contributions of the higher and
lower frequency components, in comparison to the energy associated with 0.84 Hz, are
minor. The secondary analysis, with a finer resolution, indicates that the ground motion is
not so periodic. As shown in Table 5.9, the energy in the second mode at 0.81 Hz
dominates, though there is noteworthy energy at 0.55 Hz (21.0%) and 2.38 Hz (16.4%).
TABLE 5.9
RELATIVE CONTRIBUTIONS OF EACH COMPONENT TO
INSTANTANEOUS SPECTRA FOR KOBE EARTHQUAKE PRIMARY
AND SECONDARY ANALYSES
tj
[s]
IF1
[Hz]
Eˆ1 (t j )
3.0
4.5
7.2
13.0
0.52
4.0
0.55
0.38
41.1
3.17
3.0
4.5
7.2
13.0
0.55
0.55
0.55
0.55
10.9
10.9
29.1
21.0
[%]
IF2
[Hz]
Eˆ 2 (t j )
[%]
Primary Analysis
0.84
28.9
0.80
41.2
0.91
34.6
0.84
67.7
Secondary Analysis
0.84
27.6
0.88
29.2
0.88
41.3
0.81
42.9
IF3
[Hz]
Eˆ 3 (t j )
2.60
1.80
2.38
2.24
43.2
53.1
12.5
24.7
2.67
2.67
2.31
2.38
22.2
14.3
16.6
16.4
*Broadband energy contribution centered or peaking at this frequency.
216
[%]
5.4.6 Comparison of Earthquake Properties
A comparison of the properties of the various nearfield earthquake records can be
garnered from the wavelet energy accumulation rates. As the primary analysis had
superior time resolution, its results are used to create the comparative energy
accumulation in the time domain in Figure 5.42. Conversely, the secondary analysis is
used to generate the same figure for the frequency domain. The richness of the Guerrero
MC nearfield event is quite apparent, as its accumulation of energy is very gradual over a
wide range of frequencies. The El Centro and Northridge events parallel each other, with
very similar distributions of energy with frequency, though El Centro has elevated energy
levels for frequencies greater than 3 Hz. The black horizontal lines delineate the 90 and
95% energy levels. The frequencies corresponding to these levels are tabulated in Table
5.10. As the table demonstrates, Loma Prieta comparatively has the least low frequency
energy, evident from its curve’s steep ascent followed by only modest energy gains under
2 Hz. Interestingly, the Guerrero MC nearfield record, similarly reflects a minimal energy
contribution under 1 Hz, though this minor contribution was markedly amplified in the
farfield record from the Mexico City valley, as shown previously in Figure 5.21. The
Kobe record here provides the most dramatic energy profile in the frequency domain,
emphasizing its concentration at low frequencies, a fact affirmed in Table 5.10 as it has
the lowest frequencies at the 90 and 95% levels of the analyzed records.
217
E(f)
El Centro
N. Ridge
Loma Prieta
Guerrero MC
Kobe
f [Hz]
FIGURE 5.42. Comparison of energy accumulation curves in
frequency domain for each of the five nearfield earthquake records
TABLE 5.10
COMPARISON OF ENERGETIC COMPONENTS AND ENERGY
ACCUMULATION LEVELS IN TIME AND FREQUENCY FOR FIVE
EARTHQUAKE RECORDS
El Centro
Guerrero MC
Loma Prieta
Northridge
Kobe
Most Energetic
Component
f [Hz]
t [s]
1.1
4.1
0.5
10.3
1.4
2.8
0.8
4.1
0.8
4.6
90% of Signal Energy
f [Hz]
0.93
1.09
1.41
0.76
0.54
218
t [s]
26.5
33.3
7.7
12.1
13.1
95% of Signal Energy
F [Hz]
0.64
0.53
1.21
0.50
0.40
t [s]
28.1
35.9
9.4
15.2
14.4
To further enhance the discussion, Table 5.10 also tabulates the most energetic
components of these records, identified as those inducing the maximum rate of change of
energy accumulation in the frequency and time domain. It is striking to note the parallels
between Kobe and Northridge, occurring on the same day of the year and possessing very
similar energetic components. Both have very strong energetic components at 0.8 Hz.
The Fourier spectrum did detect sizeable energy in this range for both quakes, as shown
in Figure 5.31 and Figure 5.37, however it can be implied that the enhanced low
frequency resolution in the wavelet analysis was better suited to emphasize and capture
the true energetic content at this frequency. For Loma Prieta and El Centro, again the
most energetic component is consistent with the Fourier spectral representation shown
respectively in Figures 5.25 and 5.8, though identified at slightly lower frequencies,
perhaps again due to the enhanced resolution in this regime. Interestingly, the only
record, which did not find agreement between Fourier spectral amplitudes and the
wavelet energetic components listed in Table 5.10 is the Guerrero MC record. In the
Fourier spectrum in Figure 5.16, though a low frequency presence is noted, it is dwarfed
by the energy near 2 Hz. However, the wavelet analysis in Figure 5.18 detects that this
single frequency component near 0.5 Hz in the Kobe quake is the most energetic
component, though with narrow bandwidth. Eventually, the large contributions at a wide
range of frequencies in the vicinity of 2 Hz ultimately dwarf this low frequency
component in the Fourier spectrum in Figure 5.16, but the 0.5 Hz component consistently
appears in the wavelet representation, again perhaps better detected due to the enhanced
frequency resolution. Ironically, as the quake arrives in the Mexico City valley, it is this
ever present component that is amplified and leads to widespread destruction.
219
The analysis of energy accumulation in the time domain is provided by Figure
5.43. Kobe, El Centro and Northridge each release their energy at a similar rate in time,
though the intensity of El Centro trails off sooner than the other two events, having
longer duration. The energy in Loma Prieta is released most quickly of all the records, as
noted in Table 5.10. Mexico City tends to release her energy most gradually and does not
manifest a single big event indicated by a steep ascent in the energy accumulation plot,
making this earthquake very unique by conventional standards.
E(t)
El Centro
N. Ridge
Loma Prieta
Guerrero MC
Kobe
t [s]
FIGURE 5.43. Comparison of energy accumulation curves in
frequency domain for each of the five nearfield earthquake records
220
5.4.7 Seismic Response of a Building
The previous analyses have highlighted the wavelet’s ability to capture the nonstationary
characteristics of earthquakes and reveal the evolution of frequency content with time.
Similar and more physically meaningful insights for Civil Engineers can be obtained by
applying this tool for the analysis of structural responses under these highly nonstationary
events. The unique ability of wavelets to capture these nonstationarities has made it
equally useful in the simulation of structural responses to these events (Basu & Gupta,
1997; 2000). To demonstrate the ability of the wavelet to capture nonstationary
characteristics of structural response, consider the roof level acceleration of a 13-story
commercial building (Sherman Oaks), along its longitudinal (E-W) axis during the 1994
Northridge earthquake. The measured acceleration is shown in Figure 5.44a. A wavelet
analysis of the signal, with central frequency of 2 Hz and a highly refined discretization
(OF = 0.5), was conducted. The resulting marginal spectrum and its Fourier counterpart
is shown in Figure 5.44b. From this spectral representation, it appears that the response is
dominant in the first mode near 0.38 Hz, accompanied by lesser contributions from the
second mode at 1.17 and the third mode at 2.34 Hz. The irregular shape of the wavelet
marginal spectral peak in the first mode suggests the presence of nonlinearity. While the
wavelet and Fourier representations are in good agreement in identifying the modes and
their relative contributions, this detail in the fundamental mode is obscured by a
comparable Fourier analysis. The wavelet spectrum further identifies the total energy of
the system more accurately, with a standard deviation of 56.5 cm/s2 verses the Fourier
estimate of 60.0 cm/s2, in good agreement with standard deviation of the signal itself
221
a [cm/s2]
SWT(fo/f), SFFT(f)
(a)
f [Hz]
(c)
(b)
f [Hz]
f [Hz]
(d)
f [Hz]
(e)
t [s]
(f)
f [Hz]
f [Hz]
(g)
t [s]
t [s]
FIGURE 5.44. (a) Recorded structural response to Northridge ground motion,
(b) Fourier and wavelet marginal spectra; (c) scalogram; (d) WIFS; (e) zoom
of third mode ridge; (f) zoom of second mode ridge; (g) zoom of fundamental
mode ridge
222
(56.6 cm/s2). Despite capturing the energy accurately, any time-dependence has been
completely lost.
The wavelet scalogram in Figure 5.44c preserves these evolutionary features,
identifying two bursts of energy after the tenth and thirty-fifth seconds. As evident in the
time series in Figure 5.44a, only the former large-amplitude response includes the two
higher modes. An extraction of wavelet ridges produces the wavelet instantaneous
frequency spectrum in Figure 5.44d. The ridges, though fainter for the higher frequency
modes, manifest distinct fluctuations and an apparent softening of the fundamental ridge.
Figure 5.44e zooms in on the third mode, darkening the plot and showing a hardening
with the strong shaking, followed by a decrease in stiffness. The contributions of this
mode are relatively isolated and manifest only with the strong shaking of the first 15 s.
The second mode is present in both major shaking events, and when amplified and
darkened in Figure 5.44f and echoes a similar characteristic of softening, again reaching a
minimum stiffness following the major pulse near 10s. The dominant low frequency
ridge in Figure 5.44g begins at a plateau near 10 s and then softens in the subsequent 10
s, manifesting some slight hardening during the second event, which primarily affects this
lowest mode. It is apparent that by the conclusion of the shaking, this structure’s
fundamental period has softened, an indicator of some permanent damage in the
structure. Interestingly, a comparison of the structural frequencies obtained through
testing after the quake with some estimates taken before the quake may have detected this
softening. However, without a baseline for comparison, the frequencies identified during
or after the quake shed little light. On the other hand, the wavelet analysis allows the
determination of frequency throughout the shaking, so that one record alone can be used
223
to document the onset of softening. In this context, wavelet-based analysis provides an
attractive framework for structural health monitoring in the absence of such baseline data,
which may not be available for most buildings.
The energy accumulation plots in Figure 5.45 reflect that energy is strongly
accumulated near 0.347 Hz, corresponding to the first mode of the structure in its
damaged state. The accumulation in the frequency domain plateaus at three distinct
points, readily identifying the three modes and their relative contributions. The third
mode contributions are quite scarce, while the second mode contributes 10-15% of the
total energy and the first mode is responsible for nearly 70% of the total signal energy.
Accumulation in time occurs distinctly in two events, representative two periods of
activity sandwiching the lull between 20 and 30 s. The rate of change of energy
accumulation in the time domain identifies the precise onset of the two events and the
dE(t)/dt
E(f)
E(t)
a [cm/s2]
times of maximal energy influx, in this case occurring at 11.2 s.
f [Hz]
t [s]
t [s]
FIGURE 5.45. Energy accumulation in frequency domain and time domain (left) and
structural accelerations and rate of change of energy accumulation in time domain for
Sherman Oaks building under Northridge earthquake
224
A microscopic zoom of the energy distribution at distinct points within the
response time history is provided by the instantaneous spectra in Figure 5.46. The first
spectrum is associated with the plateau region in the rate of change in energy
accumulation with time. This is a region where the rate of energy accumulation held
relatively steady prior to strong shaking. At this time, all three modes appear to be
present, though the two higher modes are sharing energy over a broad range of
frequencies. The side lobes on the fundamental spectral peak indicate the presence of
additional adjacent frequencies that could indicate a nonlinearity within that mode. Note
also the surfacing of a fourth, high frequency mode. The relative energy contributions
from these modes are listed in Table 5.11. The next two spectra are associated with peaks
in that first strong shaking, denoted by the two peaks near 10 s in the rate of change in
energy accumulation plot in Figure 5.45. At 8.7 s, there is a relatively larger contribution
from frequencies in the vicinity of the second mode, diminishing the influence of the
third mode. At 11.1 s, the energy has shifted dominantly towards the first mode coupled
by a robust distribution of energy near the third mode. The last spectrum is associated
with the final significant response within the structure. As evidenced previously, this
response is dominantly in the first mode, as evidenced by Table 5.11, with a minor
second mode contribution. The more detailed investigations of energy distribution with
time, provided in the instantaneous spectra, indicate that the energy is more prominently
associated with the higher modes in the first 9 s of the event. Even at 11.1 s, despite the
surging first mode response, there is still a strong contribution in the vicinity of the third
mode. It is only in the decline in energy accumulation rates symbolized by the dramatic
fall off after 11 s in Figure 5.45 that the first mode overwhelmingly dominates the
225
SG(a,t=8.7 s)
SG(a,t=5.0 s)
SG(a,t=37.1 s)
SG(a,t=11.1 s)
f [Hz]
f [Hz]
FIGURE 5.46. Wavelet instantaneous spectra taken at
critical time steps in the Sherman Oaks building
response in the Northridge earthquake
TABLE 5.11
RELATIVE CONTRIBUTIONS OF EACH COMPONENT TO
INSTANTANEOUS SPECTRA FOR SHERMAN OAKS BUILDING IN
NORTHRIDGE EARTHQUAKE
tj
[s]
IF1
[Hz]
5.0
8.7
11.1
37.1
0.40
0.42
0.42
0.34
Eˆ1 (t j )
[%]
32.9
40.7
50.6
93.7
IF2
[Hz]
1.38
1.13
1.08
1.10
226
Eˆ 2 (t j )
[%]
28.4
44.5
17.3
5.8
IF3
[Hz]
2.28
2.50
2.20
Eˆ 3 (t j )
[%]
19.5
13.4
31.0
response. The Fourier perspective cannot reflect such significance of higher modes over
specific durations of the response.
5.5 Wind-Induced Response of a Tall Building
While relatively stiffer structures bear the brunt of most earthquakes, it is the tall, flexible
buildings that produce dynamic responses driven primarily by wind. Under the action of
wind, oscillations of high-rise buildings occur in three simultaneous directions:
alongwind, acrosswind and torsion, as shown in Figure 5.47. The alongwind motion
primarily results from the pressure fluctuations on the windward and leeward faces,
which generally follow the fluctuations in the approach flow, at least in the low
frequency range. The lateral motion, in the acrosswind direction, is introduced by
pressure fluctuations on the side faces, induced by the fluctuations in the separated shear
layers and wake flow fields. The wind-induced torsional effects result from an imbalance
in the instantaneous pressure distribution on the building surface (Simiu & Scanlan,
1996).
The action of wind on structures may be one of the best-suited areas for the
application of wavelets due to the time-dependency of the loading. As with most
environmental loads, the magnitudes of loading are indeed time-varying; however, in the
case of wind, this situation is further complicated by the fact that the direction of loading
can also manifest marked time-variations within a single event. This unique situation can
lead to rapid changes in the structural response as the alongwind and acrosswind
directions of motion evolve with time, further complicated by the intermittent role of
torsion arising from the instantaneous distributions of pressures on the structure. The
227
FIGURE 5.47. Response components for buildings
under the action of wind loading (taken from Kijewski
et al., 2001)
ability to examine the frequency content of the resultant response alongside the timevarying wind speed and direction was previously limited prior to the advent of timefrequency transformation tools. The following example demonstrates the utility of
wavelet transforms in this regard.
5.5.1 Preliminaries
A five-year full-scale monitoring program was initiated between 1973 and 1978 on an
800-foot building in Boston that had manifested some undesirable response
characteristics and failure of building envelope components (Durgin & Gilbert, 1994).
Due to legalities, more specific details of the structure cannot be disclosed here. The
extensive monitoring program included the collection of wind speed and accelerations at
8 locations in the building: with four sensors located on the 57th floor and another four at
the 35th. In particular, these accelerometers at the higher elevation detected significant
amplitude motions capable of causing occupant discomfort. The analysis of the data
228
obtained in the monitoring program is complicated by the fact that the structure is known
to have both sway fundamental frequencies and its fundamental torsional frequency very
closely aligned. Testing of the building indicated that the fundamental sway mode in the
softer direction was 7.36 s (0.136 Hz) and the torsional mode was 6.37 s (0.157 Hz). The
locations of the four accelerometers on the 57th floor are shown in Figure 5.48. The two
sensors measuring the motion parallel to the long face of the building were termed the
“NS” sensors by the original monitoring team. The other sensors capturing motions
parallel to the building’s shorter axis were termed “EW”. The sensor pairs are positioned
at opposite corners of the building plan and named based on the column they are
associated with, i.e. 2 or 59.
FIGURE 5.48. Plan view of monitored building with accelerometer locations
and orientations
229
The record analyzed here corresponds to March 10, 1974 (record id: MR10743)
and resulted in the largest levels of peak acceleration observed in the tower over the
monitoring period. Over the course of this hour-long record, the mean wind speed,
measured at 100 ft above the rooftop, was 53.7 mph, gusting to 77.3 mph, primarily out
of the west-northwest at 290°. This mean wind direction and its orientation relative to the
building’s primary axes are shown approximately in Figure 5.48.
Knowing the close proximity of the sway and torsional frequencies, it is assumed
that beating between these modes is a very probable occurrence. The first priority was to
define the wavelet central frequency necessary to separate these modes. A 3 Hz Morlet
wavelet was selected, capable of achieving this separation down to 0.012 Hz. An overlap
factor of 1 and padding factor β = 1 were used in the analysis. Due to the potential for
beating between modes, before analyzing the record, a baseline wavelet representation of
this phenomenon was established. This is accomplished using the summation of two sine
waves of identical amplitude and with fundamental frequencies of 0.13 Hz and 0.16 Hz,
representing two closely spaced modes of vibration observed previously in data taken
from this building. Two wavelet perspectives are provided in Figure 5.49. The first is the
marginal wavelet spectrum, shown by the solid lines in Figure 5.49a. Its Fourier
counterpart is shown by the dotted lines. Both representations detect a pair of spectral
peaks at 0.135 and 0.165 Hz. Investigating the time-varying behavior of this system
through wavelet instantaneous spectra at several time intervals unveils more details of the
interaction between the two waveforms. At periods of high amplitude motion, the
bandwidth of the two modes modulates and the two appear to merge into a single mode,
though the individual peaks of the two modes are still discernable. At lower amplitudes
230
3
1
5
6
(a)
SWT(fo/f), SFFT(f)
x(t)
(b)
SWT(fo/f, t=tj)
2
4
1
Time [s]
2
3
4
5
6
f [Hz]
f [Hz]
FIGURE 5.49. (a) Comparison of wavelet marginal spectrum (solid) and
Fourier spectrum (dotted) for beating sines; (b) beating sine signal and
wavelet instantaneous spectra at 6 locations in the time history
of motion, the two modes separate again into distinct yet closely-spaced modes, with the
pattern repeating itself throughout the beating phenomenon. The wavelet consistently
identifies the two peaks at 0.135 and 0.165 Hz in each instantaneous spectrum, indicating
that while bandwidths may modulate the frequencies remain intact. However, this
idealized system manifests no nonlinearities, which may further complicate the beating
phenomenon as frequencies shift with amplitude. Further, these two sines maintain the
same amplitude level throughout the simulated signal. Thus, if amplitude modulation is
also introduced, the situation is complicated as either one of the spectral peaks may dwarf
the presence of the other due to an unequal distribution of energy at that instant. As the
subsequent analysis reveals, such amplitude fluctuations, the melding of two modes in a
beating phenomenon and the presence of nonlinearity all play a role in the response of the
monitored building in Boston. Finally, in this discussion there is no need to investigate
231
the energy associated with each component, as a cursory assessment of the relative
magnitude of the wavelet coefficients will be sufficient, contrary to the statements in
Section 5.3, since the spectral components are assumed to be in very close proximity,
they have comparable bandwidths and therefore comparable amplitude reduction to
account for increased bandwidth. Thus, in the subsequent discussions, wavelet coefficient
amplitudes will be assumed to be an accurate direct indicator of the relative strength of
each of these components.
5.5.2 Global Analysis
The analysis begins by examining the wavelet marginal spectra calculated from the entire
length of the record, shown in Figure 5.50. These provide a similar perspective as that
obtained by Fourier analysis, but through the discretization scheme in Chapter 4 these
wavelet marginal spectra can provide a smoother representation with enhanced resolution
at the lower frequencies. Note that due to the discrete frequencies chosen in the analysis
the wavelet spectra will focus at the discrete frequency closest to the actual frequency of
the system. Figures 5.50b and 5.50d manifest two modes, consistent with the closely
coupled sway and torsional modes known to exist in this structure. The softer of the two
is associated with the fundamental sway mode along the NS axis, while the higher
frequency is associated with the first torsional mode. Note interestingly that in Figure
5.50a and 5.50c, the EW sway mode is oddly not present, with only dominant torsion
detected. However, an investigation of the larger data set was able to identify the EW
sway component in other records and found that the fundamental modal frequencies from
lowest to highest are: sway detected by the NS sensors, followed by sway detected by the
232
#59-NS
#59-EW
(b)
SWT(fo/f)
SWT(fo/f)
(a)
f [Hz]
f [Hz]
#02-EW
#02-NS
SWT(fo/f)
SWT(fo/f)
(c)
f [Hz]
(d)
f [Hz]
FIGURE 5.50.Wavelet marginal spectra for the four sensors at the 57th
floor of Boston building
EW sensors, and finally torsion (Brown, 2003), indicating that all three frequencies lie in
that order within a span of 0.136 to 0.157 Hz, though suggesting that the potential for
coupled response between sway and torsion may be observed in those EW sensors.
Nevertheless, the spectra associated with this event reflect that motion at the corners of
the structure is dominated by torsion, facilitated by the larger torsional arm to that
direction. In fact, even in the NS sensors, where torsional arms are shortened, torsion still
overshadows NS sway. This is a disturbing feature since torsional accelerations are the
most perceivable motions from an occupant comfort perspective.
The magnitudes of the spectra also manifest some interesting features, e.g. the
windward EW torsional peak is twice the amplitude of its leeward counterpart’s. While
233
this feature may be attributed to the inability of the floor diaphragms to sufficiently
transmit the forces over the entire structural system (comprised solely of a moment
resisting frame and core), this does not fully explain the discrepancy. On the other hand,
the structure tends to twist and sway at comparable levels at both the leeward and
windward ends of the building in the other direction, as shown in Figure 5.50b and 5.50d.
Note however that the amplitudes of the spectral peaks are slightly diminished at the
windward side of the building in Figure 5.50d, particularly in sway. The consistency
between Figure 5.50b and 5.50d affirms a torsional frequency of 0.160 Hz and NS sway
of 0.132 Hz, while the EW sway cannot be identified from the marginal spectra. The
torsional frequency is confirmed as 0.160 Hz according to Figure 5.50c; however, Figure
5.50a provides the most interesting point of discussion in this analysis. Here at the
windward face of the building the same torsional frequency that was found at #02-EW is
not observed, but rather what appears to be a softened version (0.153 Hz). Hence, these
global spectra for the EW sensors would indicate that the building is twisting at two
different frequencies. Further, it is at this windward sensor that the largest amplitudes of
motion are detected (note the spectral amplitude twice its counterpart #02-EW),
indicating that the structure is effectively fishtailing in the wind due to an eccentricity
between its mass and elastic centers. Recall also that this is the sensor set that would be
measuring the sway mode closest to the known fundamental torsional frequency. Thus
the building’s response in both amplitude and frequency vary over the building’s plan, an
intriguing feature that will be explored utilizing wavelet instantaneous information.
The analysis of this feature and its explanation can commence through a wavelet
analysis focused on distinct events within the time histories. As sensor #59-EW manifests
234
this very interesting behavior, six high amplitude events in its response were identified
and the response of the other sensors during these six events is analyzed to determine the
overall behavior of the building. The time histories recorded at each sensor are shown in
Figure 5.51, with arrows demarcating the six events analyzed herein. Event 1 generates
the highest amplitude shaking and will be discussed at great length for each of the four,
recorded channels of data. Subsequently, analyses of some of the remaining events are
provided to reinforce observations from Event 1. Only selected figures are included from
acceleration
[milli-g]
#59-EW
#59-NS
acceleration
[milli-g]
these other events for brevity.
#02-EW
#02-NS
Time [s]
Time [s]
FIGURE 5.51. Time histories of acceleration response detected at four accelerometer
locations. Events analyzed herein denoted by arrows
X
5.5.3 Event 1: 0-200 s
The images in Figure 5.50 can be viewed as global wavelet marginal spectra, as they
represent an averaged perspective over the entire time history. On the other hand, local
235
wavelet marginal spectra can give an averaged representation of isolated signal features
over particular events. The local marginal spectrum for sensor #59-EW is provided in
Figure 5.52 and helps to unveil the particular behavior at this location during the first
event shown in Figure 5.51, typified by an isolated high amplitude burst at the beginning
of the record. Though the wavelet analysis retains the same resolution in every figure
presented herein, this local marginal spectrum identifies two peaks, one at 0.153 Hz with
very broad bandwidth and the indication of second mode sharing the same spectral band,
an interesting finding considering this sensor is measuring that more likely coupled
frequency pair. The presence of a higher mode contribution at 0.460 Hz is also
noteworthy. Gurley & Kareem (1999) recommended the application of wavelets to
identify the participation of higher modes in the response of tall buildings. Wavelets
provide one of the few viable strategies to uncover such details, since the presence of
these higher modes may depend greatly on the turbulent structure of the wind and its
evolution with time. As a result, the relative contributions of each mode may suddenly
increase for the same mean wind speed due to the instantaneous changes in the
distribution of energy at different frequencies. As this is a time-varying phenomena, it
cannot be identified through classical spectral techniques.
Analyzing the instantaneous wavelet details (Figure 5.53) during this event, the
highest amplitude response occurs when the wind direction shifts and aligns with the
corner of the building (270°) commensurate with the peak wind speed during this event.
As the wind shifts back toward 290°, the response level subdues to an extent. It is just
prior to and following this highly energetic burst that the single mode response, identified
in the global marginal spectrum at 0.153 Hz, is observed, as shown in Figure 5.53. Note
236
sWT(fo/f)
#59-EW
f [Hz]
FIGURE 5.52. Local wavelet marginal spectrum
for #59-EW (Event 1)
the large bandwidth associated with this spectral peak, broadening toward the high
frequencies, suggestive of a coalescence of the two modes. The wide bandwidth of the
response in conjunction with the wind direction supports the conclusion that both EW
sway and torsional responses are sufficiently large and simultaneous in the response,
leading to a melding of spectral bandwidth in the beating phenomenon similar to the
example in Figure 5.49. This situation can be particularly energetic if both response
components are of comparable amplitude. In spectra 1-3, this strong vibration is
stimulated by the wind direction shift, proving critical for this building’s geometry within
the site-specific surroundings. Of the data collected, this wind direction shift is only
observed in this record and this is also the only time motions of this structure reach the 20
milli-g levels. As the amplitude increases in the 3rd, 4th and 5th spectra, a shift of this
spectral peak to a lower frequency of 0.148 Hz occurs. This is the characteristic softening
of frequency that has been observed in full-scale measurements, (e.g. Tamura, 1998), as
237
Wind
Speed
[mph]
Acceleration
[milli-g]
Wind
Direction
[deg]
#59-EW
1
3
2
5
4
7
8
6
Time [s]
9
11
12
10
2
3
4
5
6
SG(fo/f, t=tj)
1
7
8
9
10
11
12
Frequency [Hz]
FIGURE 5.53. Wind speed, direction and response at #59-EW (Event 1) with wavelet
instantaneous spectral analysis
238
the contacts between structural members diminishes and greater slip in joints reduces
stiffness. This phenomenon is not indicative of permanent damage. At lower amplitudes
of motion, contacts between surfaces again resume and a frequency increase is generally
observed. More importantly, as the wind direction shifts back toward 280°, the response
level diminishes and spectra 5 and 6 begin to separate into two peaks reminiscent of what
was observed globally at the NS sensors in Figure 5.50. The torsional response gradually
eases and separates toward the higher frequencies, allowing the sway to reappears as the
dominant mode.
This behavior indicates that the isolated high amplitude motions here are
facilitated by the presence of significant EW sway and torsion. Spectra 7 and 8 continue
to manifest the bi-modal response, mostly torsional in nature, again an unsettling feature
considering the significance of torsional motion in perception. As this occurs, in spectra 9
and 10, EW sway response gradually diminishes. Note the bandwidth of the dominant
torsion in spectra 11 and 12 in comparison with the first few instantaneous spectra,
further affirmation that this energetic event early in the time history was not a single
mode response, but the product of modal coupling.
The ability of the structure to respond in such a manner is certainly facilitated by
the beat phenomenon within the system and more importantly aided by the potential
nonlinearities in stiffness and damping that can shift the torsional frequency toward the
EW sway mode and markedly accentuate spectral bandwidths. However, the continuous
beating in Figure 5.49 will not be observed in this structure due to the fact that both
response components must be in equal measure and the dominance of torsion is
239
dependent on the instantaneous distributions of wind pressure on the building’s faces.
However, Event 1 demonstrates the ideal conditions to stimulate such behavior. As
discussed by Yalla & Kareem (2001), in systems with closely coupled modes the bimodal beat phenomenon is suppressed as the two modal frequencies coalesce into a
single frequency, particularly facilitated through an increase in damping. Though the
increase in damping was shown by the authors to induce this effect, it is hypothesized
that the combinatorial effect of amplitude-dependence in stiffness, accompanied by a
moderate increase in damping achieves the same effect, though this requires amplitude
levels sufficient to induce the nonlinearity. An investigation of the behavior of the #02EW sensor, measuring the same direction of motion but on the leeward side of the
building provides added insights. At the leeward face, where global spectra in Figure
5.50c previously did not detect a discernable bi-modal response, the instantaneous
wavelet spectra clearly manifest both modes prior to the high amplitude response, though
torsion dominates (Figure 5.54). These two modes occupy the same spectral band as the
single mode response shown previously in the instantaneous spectra numbered 1-4 in
Figure 5.53. During high amplitude motion, spectra 2-3 manifest some melding and
softening of torsion into the lower sway mode. At the highest amplitude of motion, the
two modes nearly coalesce as observed at the windward face of the building, but this
transformation is not ever fully realized, reinforcing the particular local features that
facilitate this behavior. As a result, full melding never occurs in spectra 4-6, as torsion
diminishes and sway starts to dominate. By the 7th spectra, the sensor is now detecting a
dominant torsional component. Recall, this is consistent with the other sensor. Moving
toward the 9th -12th spectra, small hitches in spectra 9 and 10 are the only remnant of the
240
#02-EW
3
5
7
Acceleration
[milli-g]
1
SG (fo/f, t=tj)
2
4
10 12
8
9
6
11
Time [s]
1
2
3
4
5
6
7
8
9
10
11
12
Frequency [Hz]
FIGURE 5.54. Response at #02-EW (Event 1) with wavelet instantaneous
spectral analysis
241
sway response, and a single torsional mode response is observed to occur globally in both
Figure 5.53 and 5.54, though interestingly, the building’s torsion motion appears stiffer at
the windward face than at the leeward. This again may be the localized by-product of
fishtailing due to the orientation of the core, which is not aligned with the geometric
center of the building and is shifted more towards the windward face of the building.
Note that in the later spectra in Figure 5.54 there is no clear sway component but a
widened bandwidth at the base of the spectra, though there is some trace sway at the
windward sensor, emphasizing the localized features of this energetic event.
Examining the response in this event for #59-NS in Figure 5.55, the structure
manifests two modes in spectrum 1 with NS sway dominant, but by the 2nd and 3rd
spectra, the melding of the two fundamental modes due to a softening of torsion is
evident. Further note the appearance of a higher mode of appreciable amplitude. By the
4th and 5th spectra, the melding of both modes into a single broadband peak is again
observed. The coalescence phenomenon emerges at the highest amplitude shaking,
facilitated again by torsion frequency softening and enhanced bandwidth due to damping
increase with the amplitude of motion. The broad bandwidth of the coalesced mode in the
4th spectra supports the premise of a merger between two independent modes of
vibration. Because spectra 1-4 occur at the same amplitude level, the suppression of
either mode is not the result of it being dwarfed by the other mode. Instead, the data
reflects a physical manifestation of beat phenomenon interaction. The same holds true for
the previous discussion of the EW sensors and the following discussion of the other NS
sensor during this event.
242
#59-NS
3
5
7
9
8
11
Acceleration
[milli-g]
1
2
4
6
10 12
Time [s]
2
3
4
5
6
7
8
9
10
11
12
SG(fo/f, t=tj)
1
Frequency [Hz]
FIGURE 5.55. Response at #59-NS (Event 1) with wavelet instantaneous
spectral analysis
X
243
Akin to spectrum 6 for the EW sensor at this same location, the melded modal response
soon breaks down, but in a more dramatic fashion. The simultaneous presence of both a
coalesced mode sandwiched between sway and torsion indicates the presence of all three
characteristics within the local wavelet analysis window, though beyond this point the
coalesced mode disappears leaving again the bi-modal response. Dominant torsion
response ensues in spectra 7-10, though both modes are clearly present and better
separated than in the other axis of the building. Note that the response observed here in
the later stages of the event is one-tenth of the motion observed in Figure 5.53. Note also
that at lower amplitude levels, the torsional response occurs at a higher frequency,
consistent with the observed amplitude dependence mentioned previously.
At the leeward face, the first instantaneous spectrum in Figure 5.56, in
comparison to Figure 5.55, has a comparable level of torsion motion, but diminished
sway. Again, as observed in the #02-EW sensor, the two response components are not of
comparable amplitude a melding in spectra 1-4 that never achieves the single mode
response observed at the windward side. The lack of this feature is also due to absence of
sufficient amplitude motion to soften the torsional mode and enhance the damping in the
system. Soon after, spectra 5-8 show the emergence of the sway component, though the
motion at this point is of lower amplitude. However, the sway component is only
discernable due to the low amplitude of the torsional component at this time. By spectra
9-12, the appearance of this sway component is hardly evident and a low amplitude
torsion component remains. Note that the residual torsional component is less than its
counterpart at the windward side in Figure 5.55, due to structural eccentricities. Also
comparing to Figure 5.55, these latter spectra more clearly reflect the more energetic
244
#02-NS
3
5
Acceleration
[milli-g]
1
2
4
7
9
8
6
11
10 12
SG(fo/f, t=tj)
Time [s]
1
2
3
4
5
6
7
8
9
10
11
12
Frequency [Hz]
FIGURE 5.56. Response at #02-NS (Event 1) with wavelet instantaneous
spectral analysis
245
sway response experienced in the windward corner is not present the other side of the
building.
5.5.4 Event 2: 1350-1550 s
In Figure 5.57, it is again demonstrated that under similar spectral amplitudes, during the
second noted event, the familiar coalesced response associated with high amplitude
motion is observed again at sensor #59-NS, subsequently separating back into the two
commonly identified modes. The presence of a coalesced mode at any frequency between
the predominant sway and torsional modes (in this case at 0.153 Hz) is verified by the
marked bandwidth of the spectral peak, particularly at the base and often asymmetric. As
shown here, once separation occurs, due to a reduction of either response component,
leading to lower amplitudes of motion, a decrease in damping, and increase of stiffness,
the dominant torsional component remains. Again the prevalence of torsion is concerning
and may perhaps again be attributed to an eccentricity between the geometric and elastic
centroids of the building, as a result of the core being shifted from the centerline of the
building and toward the leeward side, as shown in Figure 5.48.
5.5.5
Event 4: 2600-2800 s
Event 4 provides an opportunity to explore the effect of sway on dominantly torsional
responses. Figure 5.58 compares the instantaneous spectra for #59-EW and #02-EW
during part of this event. At the windward side of the building, the first spectrum
indicates the notable presence of sway, as observed previously. This contribution is not
present at the leeward side. In the ensuing amplification of the sway response, the torsion
246
1
#59-NS
3
Acceleration
[milli-g]
2
SG(f/fo, t=tj)
Time [s]
1
2
3
Frequency [Hz]
FIGURE 5.57. Response at #59-NS (Event 2) with wavelet instantaneous spectral
analysis
present in the windward motions converges with the sway mode in the second spectra,
resulting in a single mode response at a lower frequency and with larger bandwidth than
the torsion response detected at the leeward side of the building. A comparison of spectra
2-4 for both sensors affirms this. This is the effect of torsion beating the sway mode at the
windward side of the building, a local effect at that critical windward corner. As the
amplitude level decreases in spectra 5 and 6, both sensors are beginning to detect similar
torsional motions manifesting comparable spectral amplitudes, bandwidths and resonant
frequencies. The windward sensor continues to manifest a low frequency asymmetry due
to some residual sway component.
The WIFS provides another useful tool for macroscopic study of the response
characteristics. The WIFS for #02-NS during Event 4 is provided in Figure 5.59. The
WIFS manifests a bi-modal response in which the energy is dominantly focused with the
247
Acceleration
[milli-g]
1
#59-EW
5
2
3
6
4
SG(fo/f, t=tj)
Time [s]
1
2
3
4
5
6
Acceleration
[milli-g]
2
Frequency [Hz]
#02-EW
5
1
3
6
4
SG(fo/f, t=tj)
Time [s]
1
2
3
4
5
6
Frequency [Hz]
FIGURE 5.58. Response at #59-EW (top) and #02-EW (Event 4) with
wavelet instantaneous spectral analysis
248
torsion component from 2600-2700 s and then transitions to dominant motions in the
sway mode. Note that the higher frequency torsional mode shows far greater fluctuations
in its instantaneous frequency ridge, demonstrating an increased sensitivity to amplitude
variations than the lower frequency sway mode. During low amplitude events, e.g. 2700
–2730 s, the frequency is higher than during stronger vibrations, e.g. at 2660 s. This may
explain why the torsional mode is often observed to soften into the sway mode and
Acceleration
[milli-g]
induced the coalesced response associated with beat phenomena.
Frequency [Hz]
#59-NS
Time [s]
FIGURE 5.59. WIFS for #02-NS (Event 4)
249
5.5.6 Event 5: 3000-3200 s
Figure 5.60, spectra 1 displays a dominant torsion response of #02-EW during Event 5, a
feature often expected at this sensor. In spectra 1-4 the separation between the dominant
sway torsion and the diminished sway component is very evident. In this case, as
the amplitude of motion increases, the dominant torsion mode associated dwarfs any
sway component. Moving through spectra 5-9, the frequency of this torsional response
softens, as the sustained amplitude of building motion increases. The bandwidth here
does not indicate an expansion indicative of the coalescence observed previously. It may
instead be hypothesized that the high amplitude torsional responses in Event 5 led to a
softening of that modal frequency leading to the shift toward 0.153 Hz, a fact affirmed
through the WIFS analysis shown in Figure 5.61. The WIFS shows only trace sway and a
gradual softening of torsion frequency during the high amplitude response, after which it
begins to stiffen again. It is this increased energy in the softened torsion mode led to
heightened spectral amplitude, so much so that it completely dwarfed any sway response.
Note that this softening is not evident in the global spectrum and only apparent in the
local analysis presented here. Though omitted for brevity, the same dominant single
mode torsional softening is observed at #59-EW in high amplitude responses, meanwhile,
the NS sensors maintain the presence of both sway and torsion at low amplitudes. In this
event, the motion is clearly dominated by torsion, obscuring any contributions due to
sway. This prevalence of torsion is the infamous characteristic of this building.
250
1
Acceleration
[milli-g]
2
3
5
6
7
#02-EW
9
4
8
SG(fo/f, t=tj)
Time [s]
1
2
3
4
5
6
7
8
9
Frequency [Hz]
FIGURE 5.60. Response at #02-EW (Event 5) with wavelet
instantaneous spectral analysis
X
5.5.7 Conclusions Based on Wavelet Spectral Analysis
In conclusion, the strongest and most sudden shaking was associated with Event 1, during
which a sudden change in wind angle led oncoming winds directed toward the west
corner of the building, an orientation more conducive to separation of flow, helping to
further simulate strong torsional motions due to alternating vortices on the two long faces
251
Acceleration
[milli-g]
Frequency [Hz]
#02-EW
Time [s]
FIGURE 5.61. WIFS for #02-EW (Event 5)
of the structure. The comparable levels of sway and torsion in this event, and the high
amplitudes of motion, facilitated a softening of the torsion mode and enhanced the
damping sufficiently such that the beating of the two coalesced into a single mode
response of high amplitude wide spectral bandwidth. As the wind direction changed
quickly from this orientation, the isolated burst in response subsided, and the modes
separated again, as the torsional component inducing beating diminished and
nonlinearities in stiffness and damping were less pronounced. The coalescence associated
with beat phenomenon only occurs under unique situations where the amplitude of
motion is high enough to enhance nonlinearities and when torsional and sway
contributions are significant. However, this phenomenon appears to be highly local, since
252
this coalescent phenomenon was only observed at the windward side of the building as a
result of a fishtailing spurred by structural eccentricity. Further, since the NS sway
frequency its orthogonal counterpart, the potential for coupling with the torsional mode is
diminished, explaining the lack of coalescent behavior in these sensors.
Further, the potential for coupling of the sway and torsional modes often
eliminates any physical decoupling of sway and torsion through a simple algebraic
combination of the sensor outputs. It is important to note that the manifestation of
amplitude dependence in dynamic properties and the evolution of a coalescent beat
phenomenon are not apparent from just a global spectral representation, such as that of
Figure 5.50. It is only in the in-depth investigations of time-frequency energy evolution
that such interaction between modes can be uncovered to enhance understanding of
building performance. Fortunately, these investigations of instantaneous modal properties
are now made possible through the wavelet framework presented here.
5.6 Offshore Platforms
The previous analysis of wind and seismic response in buildings highlighted the ability of
wavelets to uncover nonlinear and intermittent response characteristics. The final analysis
in this chapter, focused on the response of tension leg platforms (TLP) under the action of
wind and waves, will demonstrate the ability of the transform to capture time-varying low
frequency content within the resulting response. The TLP response analyzed herein was
obtained from experiments conducted through a joint industry project discussed in Gurley
(1993). This experiment measured the response of a 1:160 scale model in six degrees of
freedom, as shown in Figure 5.62, with longitudinal, transverse and vertical motions,
253
FIGURE 5.62. Schematic of response
components for tension leg platforms (taken
from Kareem, 1985)
referred to as surge, sway and heave, respectively, and angular motions in a transverse,
longitudinal and horizontal plane: respectively, roll, pitch and yaw.
The thrust toward deep-water drilling gave birth to this class of compliant
structure, designed to move with the environmental loadings rather than resisting them
rigidly (e.g. Kareem, 187). One benefit of compliance is the ability to design these
platforms with very low natural frequencies in surge, sway and yaw degrees of freedom,
as shown in Figure 5.1. As wave motions have a narrow spectra centered about
comparatively higher frequencies, compliant structures generally will not experience any
dynamic amplification of wave-induced response, apart from second-order effects.
However, they are relatively more susceptible to dynamic effects of wind than
conventional fixed leg platforms and therefore must be designed to resist both dynamic
254
wind and wave loadings (Kareem, 1987). As these loading sources simultaneously
produce low frequency wind and wave excitations, as well as higher frequency wave
excitations, this situation is well-suited for a multi-resolution wavelet analysis, exploiting
the wavelet’s low-frequency resolution capabilities and the wavelet instantaneous
spectral representations to distinguish the response components associated with wind and
those resulting from wave action. Further, the ability of wavelets to track nonlinear
features, as demonstrated in the previous analysis of building response, will have added
utility in the analysis in light of the known system nonlinearities and those of the wave
field (Kareem, 1985). Since the heave, pitch and roll are beyond the wave spectrum, they
will not be considered at this time.
Before beginning the analysis of the platform response under the combined
effects of wind and wave loads, some details of the oncoming wave field and the
experimental model are defined. First, all discussions that follow will consider the scaled
experimental data to give an indication of the likely behavior of this system at full-scale.
Two sea states will be considered in this section, the first a mild sea state with significant
wave height of 9.61 m and peak frequency, as determined from the Fourier power
spectrum in Figure 5.63a, of 0.077 Hz. The severe sea state, whose Fourier power
spectrum of wave height is provided in Figure 5.63b, is characterized by a wave
frequency of 0.056 Hz and significant wave height of 20.23 m. Again, additional
information on this experimental program, including the properties of the platform model
and the experimental facilities, can be found in Gurley (1993).
255
(b)
SFFT(f)
SFFT(f)
(a)
f [Hz]
f [Hz]
SG(fo/f, t=tj)
(d)
SG(fo/f, t=tj)
(a)
(e)
SG(fo/f, t=tj)
surge
[m]
FIGURE 5.63. Fourier power spectra for wave field in (a) mild sea
state and (b) severe sea state
(f)
f [Hz]
(b)
(g)
SG(fo/f, t=tj)
f [Hz]
(c)
f [Hz]
t [s]
FIGURE 5.64. Severe sea state (a) surge response, (b) scalogram, (c) WIFS and
instantaneous spectra at tj = (d) 1007.8 s, (e) 1542.2 s, (f) 1970.3 s, (g) 2155.0 s
X
256
The surge response (Figure 5.64a) is analyzed first, using a wavelet analysis with fo = 0.5
Hz to unveil any time-varying characteristics associated with wave loading. Considering
that typical wave spectra are known to peak near 0.08 Hz and wind spectra at generally
lower frequencies, the wave-induced response of the platform can be readily
distinguished through the wavelet analysis. Both a mild and severe sea state are analyzed
and compared herein to emphasize the relative role of wave action in each. The
scalogram in Figure 5.64b identifies the wave frequency response with added intermittent
components, which can be attributed to occasional slamming of waves on the platform
hull in the severe sea state through the light, higher frequency bands. These oscillations
of frequency are also observed later in Chapter 7 in the discussion of random sea states
characterized by nonlinear waves. The precision of the WIFS in Figure 5.64c indicates
that this wave component within the severe sea state manifests significant fluctuations in
the instantaneous frequency. Also noted are the gradual fluctuations in the ridge
associated with the resonant response of the platform near 0.007 Hz, manifesting
nonlinearities consistent with the modeling of stiffness as a hard spring in the scaled
model, increasing with displacement. Due to the long period of response, this occurs very
gradually. Note the increase in instantaneous frequency between 800 and 1400 s followed
by a softening between 1400-2000 s and a recurrent hardening beyond 2000 s.
Meanwhile, the fluctuations in the wave component near 0.06 Hz follow the trends in the
amplitude modulation of the time history. This is perhaps most clearly indicated by the
two modulations in the WIFS between 2100 and 2400 s, following amplitude
modulations of the same form in 5.64a. The low frequency ridges may also represent
contributions from the second order, slowly varying drift forces at low frequencies,
257
caused by the second-order contribution resulting from interactions between components
of the wave spectrum.
The WIFS demonstrates that the occurrence of peak surge response is not always
commensurate with the wave frequency response, as large response excursions may also
result from low frequency hydrodynamic loads due to the passage of large waves. For
example near 1200s, a large surge response occurs without a strong wave frequency
response component. This unique wavelet perspective allows differentiation of the
relative contributions of wind and waves, both at the wave frequency and at lower
frequencies, to a given event, readily garnered through examinations of the instantaneous
spectra in Figure 5.64d-g. For example, Figure 5.64e has a significant high frequency
contribution associated with wave frequency response while Figure 5.64g has a
significant low frequency component. Comparing Figure 5.64d and 5.64e, it becomes
evident that high amplitude surge response is not always driven by higher frequency
wave frequency response and further distinguishes response due to higher frequency
wave frequency response and low frequency wind/wave effects.
On the other hand, in the mild surge response, wind forces create a large response
at the resonant frequency of the platform. The differences in the response characteristics
between the mild and severe sea states are readily summarized by comparing their
Fourier spectra and wavelet marginal spectra in Figure 5.65a. Both responses feature a
low frequency component in the vicinity of the system’s natural frequency, which arises
due to nonlinear interactions between the inputs, resulting in forcing energy in a wider
range of frequencies (Kareem et al., 1998). The wavelet marginal spectra are shown as
258
(b)
E(f)
SWT(fo/f), SFT(f)
(a)
f [Hz]
f [Hz]
FIGURE 5.65. Comparison of surge response in mild (blue) and severe
(red) sea states: (a) wavelet marginal spectrum (solid) and Fourier
spectrum (dotted); (b) energy accumulation with frequency
X
solid lines and their Fourier counterparts are shown as dotted lines. The enhanced low
frequency resolution capabilities of the wavelet become immediately apparent. In the
severe sea state (red), the component associated with the high frequency wave excitations
is immediately apparent. Though, as shown by the energy accumulation plots in Figure
5.65b, this wave component, while more pronounced, contributes less than 10% to the
overall response: the severe sea state surge response is still dominated by the energetic
low frequency wind and second order wave effects, as shown by the dark red hues in the
scalogram of Figure 5.64b. The majority of energy in this case is associated with
frequencies less than 0.01 Hz.
259
In the mild sea state, the energy concentrates at decidedly higher frequencies than
the severe sea state, as shown again in Figure 5.65b. The mild sea state surge response
(blue) has a richer distribution of energy in the frequency band under 0.02 Hz, where
98% of the energy is concentrated. Note the absence of energy at higher frequencies as
further evidence of a lack of sizeable wave frequency response in the mild sea state. As
shown by the scalogram in Figure 5.66b, the high frequency wave frequency response is
minimal, though again manifesting an intermittent, time-varying frequency character. The
WIFS in Figure 5.66c demonstrates the interchange of energy between second order
wave components and resonant wind response. The hitches in these low frequency ridges
in the vicinity of high amplitude events at 1800 and 2000 s may be due to hardening
characteristic of the platform resonant response. In particular, the mild sea state surge
analysis demonstrates that the primary effect of waves is not significant to the overall
motion of TLPs, though it is well known that slowly varying drift forces at low
frequencies caused by nonlinear interactions between wave components in wave
spectrum may result in low frequency resonant oscillations of TLPs (Kareem, 1985). The
prevalence of these low frequency components is highlighted by the instantaneous
spectra in Figure 5.66d-g.
Ideally in the case of wind and waves approaching at a zero angle of attack, there
should be no sway motion. However, due to the coupled nature of the TLP’s degrees of
freedom, the lateral motion may be induced by vortex shedding or through coupling that
exists with the yaw motion. Sway may also result from any misalignment of the model in
the wind/wave test facility. The resulting sway response in the mild sea state, garnered
from a fo = 1 Hz wavelet analysis, is characterized by a significant component of energy
260
(d)
SG(fo/f, t=tj)
surge
[m]
(a)
SG(fo/f, t=tj)
SG(fo/f, t=tj)
f [Hz]
(c)
(e)
(f)
SG(fo/f, t=tj)
f [Hz]
(b)
(g)
f [Hz]
t [s]
FIGURE 5.66. Mild sea state (a) surge response, (b) scalogram, (c) WIFS and
instantaneous spectra at tj = (d) 891.6 s, (e) 1168.0 s, (f) 1459.1 s, (g) 1772.2 s
in the wavelet marginal spectrum (blue) associated with the wave frequency response, as
denoted by the energetic contribution near the peak of the wave spectrum in Figure 5.67a.
This component contributes 30% of the total energy to the sway response, as shown by
the accumulation curve in Figure 5.67b, a comparatively greater contribution that than
observed in the surge response. As shown by the scalogram in Figure 5.68b and its
associated WIFS in Figure 5.68c, this component of the response is highly energetic and
261
(b)
E(f)
SWT(fo/f), SFT(f)
(a)
f [Hz]
f [Hz]
FIGURE 5.67. Comparison of sway response in mild (blue) and severe (red) sea
states: (a) wavelet marginal spectrum (solid) and Fourier spectrum (dotted); (b)
energy accumulation with frequency
X
shows marked fluctuations, similar to what was observed in the surge response of the
severe sea state, a detail obscured in a conventional Fourier analysis.
Conversely, the severe sea state sway response has a strong, quasi-static
component and the absence of any wave frequency response, a feature similarly affirmed
by the energy accumulation plot in Figure 5.67b. This low-frequency component
manifests trends consistent with that observed in the surge response. The scalogram
associated with this sea state is shown in Figure 5.69. Note the increasing prominence of
the resonant component of the response and the decreasing frequency of this response as
the amplitude of sway declines beyond 1600 s, again suggesting the manifestation of the
hardening spring action in the scaled model.
262
sway [m]
(a)
f [Hz]
(b)
f [Hz]
(c)
t [s]
FIUGRE 5.68. Mild sea state (a) sway response, (b) scalogram and (c) WIFS
X
X
263
E(f)
SWT(fo/f), SFT(f)
sway [m]
f [Hz]
(b)
(a)
(a)
(b)
f [Hz]
f [Hz]
FIGURE 5.70. Comparison of yaw-induced lateral response in mild (blue) and
severe (red) sea states: (a) wavelet marginal spectrum (solid) and Fourier
spectrum (dotted); (b) energy accumulation with frequency
t [s]
FIGURE 5.69. Severe sea state (a) sway response, (b) scalogram
The lateral motion induced by yaw is the final response component considered here using
a fo = 1 Hz wavelet analysis. This response component arises from asymmetrical
distributions of the wind or wave loads but can be accentuated by uneven pre-tension in
the tendons and potential eccentricity in the center of mass, which can induce surge-yaw
coupling (Gurley, 1993). As shown by Figure 5.70b, the accumulation of energy in both
the mild (blue) and severe (red) sea states is nearly identical, concentrated entirely in the
low frequencies under 0.02 Hz. The difference lies in how this energy distributes within
the low frequency range. As shown by Figure 5.70a both the wavelet (solid) and Fourier
(dotted) spectra, the mild case manifests a sharp peak with a broad bandwidth and antisymmetric shape. This spectral peak is flanked to the right by a slight, quasi-static
264
(b)
E(f)
SWT(fo/f), SFT(f)
(a)
f [Hz]
f [Hz]
FIGURE 5.70. Comparison of yaw-induced lateral response in mild (blue) and
severe (red) sea states: (a) wavelet marginal spectrum (solid) and Fourier
spectrum (dotted); (b) energy accumulation with frequency
component. The severe sea state also shows this same quasi-static component but of a far
more pronounced stature, however, the single response peak observed in the mild sea
state has separated into a bi-modal characteristic, retaining the same resonant frequency
as the mild state but now dwarfed by a higher-frequency component previously buried in
the widened bandwidth of the mild sea state.
The averaged representation provided by the marginal spectrum fails to shed light
on the true behavior of the yaw response. The scalogram in Figure 5.71b indicates that
this bi-modal characteristic is present intermittently throughout the mild sea state time
series in Figure 5.71a. The marginal spectrum merely represented the occasional presence
of the higher mode in this pair as a widened bandwidth. The presence of both modes is
very pronounced between 800 and 1200 s, concentrating at near 0.01 and 0.005 Hz.
265
yaw-induced
sway [m]
(a)
yaw-induced
sway [m]
f [Hz]
(b)
(c)
f [Hz]
(d)
t [s]
FIGURE 5.71. (a) Yaw-induced lateral response for mild sea state and (b)
scalogram; (c) yaw-induced sway response for severe sea state and (d)
scalogram
266
Between 1200 and 1400 s it reverts strongly back toward a resonant response at 0.005 Hz
only revert back to this bimodal characteristic at 1400 s, repeating this trend later in the
scalogram. The trace presence of a quasi-static component near 0.002 Hz is evident
throughout though most pronounced between 1400 and 1800 s. The severe sea state
produces a yaw scalogram in Figure 5.71c that demonstrates the enhanced presence of the
higher mode in that bi-modal characteristic near 0.013 Hz, particularly between 800 and
1000 s and from 1800 s until the termination of the time series. Strong periodicity in the
time series indicative of resonant response is identified by two isolated bursts of bright
hues in the scalogram at 1200 and 2000 s. The quasi-static response is also very strongly
evident in the severe sea state, becoming far more pronounced and persistent from 1000 s
and on.
5.7 Summary
This chapter presented specific examples of wavelet analyses of environmental loads and
associated response of Civil Engineering structures, highlighting the wavelet’s ability to
uncover intermittent and nonlinear characteristics through wavelet scalograms and
instantaneous frequency spectra. In particular, the ability to detect and model nonlinear
characteristics is particularly important for seismic analysis, where large amplitude
response and structural damage can induce frequency variations. Further, in the design
and modeling of compliant structures, wavelets can be beneficial in distinguishing wind
and wave driven response of systems with nonlinearities in stiffness and subject to the
action of nonlinear wave fields. In fact, this inherent nonlinearity in wave excitations will
be a subject of further study in Chapter 7, where wavelets will be used in the analysis of
267
Stokian waves and to uncover the intermittent characteristics of random sea states.
However, even under the lower amplitude excitations of wind, wavelets were shown in
this chapter to be a critical tool in unlocking peculiar behavior trends in a tall building
due to intermittent aerodynamic actions that were obscured in Fourier analyses. The
following chapter will further demonstrate the utility of wavelets in unlocking such
intermittent characteristics in wind fields through a wavelet-based coherence measure and
its associated processing tools.
268
CHAPTER 6
WAVELET SYSTEM IDENTIFICATION II:
APPLICATIONS TO WAVELET COHERENCE
6.1 Introduction
As motivated in the preceding chapters, though transients have long defined signature
characteristics across the engineering spectrum, available analysis tools like Fourier
transforms have been ill equipped to represent this phenomenon. It was not until the timefrequency revolution of the 20th century that such signals could be adequately treated by
the likes of the Gabor transform and the Wigner-Ville distribution. Today, wavelet
transforms lead the transition to this new analysis domain, providing the ability to display
time and frequency information independently and unveil the hidden features of
evolutionary phenomena.
Particularly in the areas of aerodynamics and wind engineering, wind field
fluctuations result in spatio-temporal pressure fluctuations on the surfaces of bluff bodies,
e.g. buildings. These pressure fluctuations are manifestations of complex, nonlinear
interactions that take place as the wind passes around a bluff structure. The spatiotemporal pressure fluctuations exhibit drastic transient features depending on their
269
location on the surface and, with the exception of the windward face, are not amenable to
a functional relationship with the oncoming wind field. Efforts to identify significant
linear correlation between wind and pressure fluctuations were unsuccessful, especially
in the separated flow regions. This has led to the consideration of higher-order
correlation, e.g. bicoherence (Gurley et al., 1997a). However, these efforts highlighted
the inability of such Fourier-based measures to capture transient higher-order correlations
that may exist between wind and pressure fluctuations.
With the availability of time-frequency analysis via wavelets, linear correlation
analyses were enhanced by way of the coscalogram (Gurley & Kareem, 1999). This
approach did identify some intermittent correlation between wind and pressure and will
be further developed in this chapter as a tool for delineating any previously obscured
intermittent relationship between certain wavelengths in the approach flow and the
resulting pressure fluctuations. The potential for such insights have made wavelets an
insightful tool for other applications in wind engineering. For example, early
investigations of turbulent wind effects were conducted by Farge (1992), who applied
wavelet-based spectral analysis to the modeling of atmospheric turbulence. Gurley &
Kareem (1997a) later adapted this to the analysis of turbulence and resulting pressures in
full-scale dynamic response data. In total, the use of wavelet transforms in this field
continues to advance, as overviewed by Gurley & Kareem (1999) and applications of
wavelets in wind, offshore, and earthquake engineering in Chapter 5.
This chapter continues the work in wavelets for both wind and waves by applying
wavelet transforms to identify first-order intermittent correlation between measured
270
records. While this representation allows a display in terms of time and frequency, the
influence of noise in the estimation of coherence over a localized time frame is
significant, making a distinction between the true correlation and noise a major issue to
be addressed. This chapter revisits the classical approach for reduction of variance,
averaging, in the multi-resolution context of wavelets, and later discusses three tiers of
denoising schemes, which minimize the need for localized averaging in order to preserve
temporal information. While hard thresholding based on global maxima of the wavelet
coherence map can be used to isolate meaningful coherence, other thresholding
simulation schemes are proposed to provide a reference noise map to more accurately
separate spurious noise effects from true signal content. These reference maps are
generated using independent realizations of time histories to establish a statistical
measure of the expected noise in the estimated coherence. The robustness of these
schemes is established using both simulated and measured data. The method is shown to
significantly reduce the presence of spurious coherence, even in cases where variance and
leakage are prevalent.
6.2 Wavelet Coherence Background: Scalogram and Coscalogram
The localized wavelet coefficients are well suited for analyzing non-stationary events,
with their squared values plotted on a time-scale (time-frequency) grid. This
visualization, called the scalogram or mean square map, reveals the frequency content of
the signal at each time step to pinpoint the occurrence of transients while tracking
evolutionary phenomena in both time and frequency. However, in some recent studies,
the concept of the scalogram has been advanced to identify correlation between signals in
271
which the squared coefficients are replaced with the product of the wavelet coefficients
of two different processes (e.g., Hudgins et al., 1993; Gurley & Kareem, 1999). This
coscalogram produces a view of the coincident events between the processes, revealing
time-varying pockets of correlation over frequency.
To demonstrate this concept, full-scale pressures measured on a building and the
upstream wind velocity fluctuations were analyzed by Kareem & Gurley (1999). The
scalogram of wind velocity and simultaneously measured pressure are presented along
with their coscalogram in Figure 6.1a-c. The light hues of the coscalogram identify areas
of correlation. Figure 6.1d-f presents the same information for two uncorrelated records.
The resulting coscalogram (Fig. 6.1f) of these two unrelated processes shows no distinct
correlation. The coscalogram contains wavelet coefficients determined from segments of
the signal isolated by the sliding window of the scaled parent wavelet. At each time step,
the calculated wavelet coefficients comprise a single raw spectrum across the range of
scales, equivalent to a spectrum obtained from a single time history in the traditional
Fourier analysis. These raw spectra that are assembled along the time axis in the
scalogram and coscalogram lack the ensemble averaging necessary in traditional Fourier
methods to reduce the variance in the estimate, resulting in noisy displays where
correlated events are difficult to differentiate from random coincident coefficients.
Though this simple measure of correlation has been used to qualitatively identify firstorder wind velocity and pressure relationships (Gurley & Kareem, 1999), it is refined in
this chapter by the introduction of a wavelet coherence measure.
272
(d)
Scale, a
Scale, a
(a)
(b)
W(a,t)
Scale, a
Scale, a
(e)
(c)
W(a,t)
Scale, a
Scale, a
(f)
Time [s]
Time [s]
W(a,t)
FIGURE 6.1. (a) Scalogram of upstream wind velocity 1; (b) scalogram of rooftop
pressure; (c) coscalogam of these two correlated processes; (d) scalogram of upstream
wind velocity 2; (e) scalogram of rooftop pressure; (f) coscalogram of these two
uncorrelated processes (taken from Gurley & Kareem, 1999)
6.3 Wavelet-based Coherence Map
As the Morlet wavelet introduced in Chapter 4 is merely a localized form of the Fourier
transform, it can intuitively be substituted into classical spectral measures to uncover
time-varying frequency content, effectively windowing the Fourier analysis. For this
reason, it is again the preferred wavelet in this application. The equivalence of traditional
Fourier measures with those newly recast using Morlet wavelets was previously shown in
Gurley & Kareem (1999) for quantities such as scalogram and coscalogram, analogs to
273
the auto-spectrum and cross-spectrum. Their abilities to capture marginal spectral
characteristics and measures of signal energy were further validated in Chapter 5. In this
chapter, the classical coherence definition is modified utilizing spectra defined locally by
Morlet wavelets to yield a time-frequency coherence function. The traditional form of the
coherence function can be retained as the ratio of the wavelet cross-spectrum to the
product of the wavelet auto-spectra of the two signals x(t) and y(t). The wavelet
coherence map is thus defined as
(c (a, t ))
W
2
=
W
(a, t )
S xy
2
W
(a, t )S Wyy (a, t )
S xx
,
(6.1)
where the localized power spectra discussed above are given by
SijW (a, t ) = ∫ Wi * (a, t )W j (a, t )dτ
(6.2)
T
with the subscripts on the wavelet coefficients indicating their association with each
signal.
The localized time integration window in Equation 6.2, T = [t − ∆T , t + ∆T ] , is
selected based on the time resolution desired in the resulting coherence map and
essentially performs the same ensemble averaging operation, albeit localized in time, as
traditional Fourier analysis to obtain an auto-spectrum or cross-spectrum of two signals.
The map is bounded between 0 and 1 and provides a view of the localized correlation
with respect to both time and frequency. Equivalence of this proposed time-frequency
coherence map with its classical formulation is demonstrated in the following section.
274
It should be noted that discussions in Torrence & Compo (1998) highlight that an
earlier coherence measure defined by Liu (1994), similar to that used by Shin et al.
(1999), had limited physical meaning without the smoothing introduced here by T in
Equation 6.2, and hint that averaging to some extent is necessary to provide a useful
measure of coherence. The parameter T in Equation 6.2 addresses this concern, although,
being somewhat arbitrary, it also presents the potential for a loss in time localization.
6.3.1 Comparison of Wavelet- and Fourier-based Coherence Estimates
The validity of the coherence map in Equations 6.1-6.2 is demonstrated by first applying
the wavelet-based coherence to stationary signals. The standard Fourier-based coherence
estimate is directly compared with the wavelet-based coherence by averaging out the time
information in the wavelet coherence map, according to
(c
W
(a ))2 =
1
nt
nt
∑ (c(a, t ))
i
2
,
(6.3)
i =1
where nt is the number of discrete time steps resulting from the localized time window, T.
The signals being analyzed in this example are the upstream wave elevation and
the resulting surge response of a tension leg offshore platform (TLP) 1:200 scale model,
measured experimentally in a wind/wave tank facility and shown in Figure 6.2a,b.
Sampled at 1 Hz, 4096 s of data is used in this analysis. This is a part of the larger data
set investigated in Section 5.6. Standard Fourier coherence estimation and Equations 6.16.3 are applied to these signals with the results shown in Figure 6.2c. The coherence is
well represented by both estimates, demonstrating the accuracy of the wavelet-based
275
Wave Elevation
[m]
(a)
TLP Response [m]
Time [s]
(b)
Time [s]
(c)
Coherence
FFT
Wavelet
Frequency [Hz]
Coherence
(d)
Frequency [Hz]
FIGURE 6.2. (a) Incoming wave surface elevation; (b)
TLP surge response; (c) wavelet and FFT coherence
estimates between wave elevation and TLP response; (d)
wavelet and FFT coherence estimates between wave
elevation and TLP response with incoherent nose added to
each
276
coherence estimate with respect to both magnitude and frequency. A second example
demonstrates that wavelet-based coherence can accurately estimate smaller levels of
linear correlation. In this case, independent white noise vectors are added to the wave
surface and TLP response time histories to reduce the level of correlation, and coherence
estimates are again produced. Figure 6.2d shows the wavelet coherence representing the
reduced correlation accurately.
6.3.2 Application of Wavelet-Coherence to Non-stationary Signals
The previous section demonstrated that wavelet-coherence viewed only with respect to
frequency provides an effective coherence estimate. The advantage of wavelet-based
coherence in identifying the time at which coherence between two signals occurs is now
demonstrated by its application to velocity and pressure signals with known pockets of
short duration correlation. Two independent Gaussian wind velocity signals (v1(t), v2(t))
are simulated for 2048 s at 1 Hz. A pressure record is then created by combining
independent white noise ε (t) with the v2(t) wind record and a quadratic term
pr (t ) = ε (t ) + v 2 (t ) + G (ε (t )2 + v 2 (t ) 2 ) .
(6.4)
For this example, G = 0.05. Two small segments of the pressure record, over the time
interval t’, are then replaced with signals generated by
pr (t ' ) = ε (t ' ) + v1 f (t ' ) + G (ε (t ') + v1 f (t ' ) 2 ) ,
2
277
(6.5)
where v1f(t') indicates v1(t’) after band pass filtering is applied to correlate the pressure
and the velocity record over selected frequency ranges. The use of Equation 6.5 produces
a pair of signals, pr(t) and v1(t), correlated only from 512 to 768 s between 0.0625 and
0.25 Hz and from 1536 to 1792 s between 0.19 and 0.37 Hz, and uncorrelated
everywhere else.
The standard Fourier-based coherence between v2(t) and pr(t) and between v1(t)
and pr(t) are displayed in Figure 6.3. Note that the intermittent coherence between v1(t)
and pr(t) cannot clearly be distinguished, suggesting that the signals are uncorrelated,
whereas the relationship between v2(t) and pr(t), as expected, appears fairly strong.
0.7
v1 & pr
v2 & pr
0.6
Coherence
0.5
0.4
0.3
0.2
0.1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Frequency [Hz]
FIGURE 6.3. Coherence between v2(t) and pr(t) and between
v1(t) and pr(t)
278
Isometric and overhead views of the wavelet-coherence map between v1(t) and
pr(t), as generated by Equation 6.1, are shown in Figure 6.4. For this example, and those
which follow, a value of fo = 5/(2π) was deemed sufficient to provide the necessary time
and frequency resolutions, though a more precise frequency resolution (larger fo) may be
required in other cases, as discussed in more specifically in Chapter 7. A time integration
window of T = [t - 64 s, t + 64 s] was applied. Pockets of strong correlation can then be
identified in these displays that include the time and frequency regions of introduced
correlation, approximately boxed. However, the coherence estimate also displays
phantom correlation in regions where no correlation exists, particularly in the low
frequency regions as emphasized by the semi-logarithmic plot in Figure 6.4. This noise is
similar to that seen in Fourier-based spectral methods, where, due to a finite number of
ensembles, variance errors are introduced. In the case of the wavelet coherence map, the
localized time integration window T determines the number of ensembles used in the
estimation of coherence in Equation 6.1. Increasing T can reduce the noise in the
coherence estimate at the expense of temporal resolution.
6.4 Minimization of Spurious Coherence
Classically, the presence of variance in raw spectral estimates necessitates the use of
averaging in order to obtain more reliable results. However, the transient information
sought in a time-frequency analysis may be obscured through excessive averaging,
especially in the low frequency regime, where spurious coherence seems most prevalent.
A variable integration scheme is proposed in the following section to address this issue,
279
Wx(a,t)
Frequency
[Hz]
Time [s]
Frequency [Hz]
log(Frequency) [Hz]
cW(a,t)
Time [s]
FIGURE 6.4. Wavelet coherence map between v1(t) and pr(t),
also shown as semi-logarithmic in frequency, emphasizing low
frequency content
280
followed by alternative approaches designed to better preserve temporal resolution using
simulated signals to build reference noise maps.
6.4.1 Multi-Resolution Integration Windows
In the initial formulation of the wavelet coherence, the localized time window is constant
throughout the analysis. However, unlike its Fourier counterpart, the wavelet transform is
multi-resolution, having scale-dependent time and frequency resolutions. Each wavelet
coefficient, at given (frequency) scale ai and time tj, is the result of analyzing a local
section of the time history windowed by the scaled Gaussian function of the Morlet
wavelet. Thus, the signal’s content ∆ti before that point in time and ∆ti after that point in
time is used to estimate the wavelet coefficient W(ai,tj), where ∆ti is dictated by ai
according to Equation 4.5. Note that the central frequency implicitly dictates this
resolution and becomes one of the parameters influencing the extent of the spurious
coherence in the wavelet coherence map. This is demonstrated in Figure 6.5 using the
simulated velocity and pressure signals with known coherent pockets boxed in the figure.
In this demonstration, a constant 64-second integration window was used to evaluate the
coherence every 10 seconds to expedite calculations. The larger values of central
frequency elongate the temporal windows of the scaled wavelets, reducing the averaging
achievable in the analysis and leading to extensive spurious coherence. The spurious
coherence diminishes for lower wavelet central frequencies as more averaging is
achieved, but this also implies that the frequency resolution capabilities are lost, leading
to a smearing of energy. A balance between the two extremes is achieved for fo=5/2π and
thus this is why it is used as the baseline case in this discussion. The influence of central
281
fo=5/2π Hz
Frequency [Hz]
Frequency [Hz]
fo=5 Hz
Time [s]
Time [s]
Frequency [Hz]
fo=1/π Hz
Time [s]
cW(a,t)
FIGURE 6.5. Influence of central frequency on the occurrence of
spurious coherence in wavelet coherence map. Known coherent
pockets between v1(t) and pr(t) designated by boxes
282
frequency in the context of the developments in Chapter 4 should be kept in mind
throughout the subsequent discussions.
For the analysis conducted in Section 6.3.2 and shown in Figure 6.4 (referred to
herein as the baseline example), the choice of a constant window spanning a total of 128
s implies that at very low frequencies, as little as one unique local section of the time
history is effectively being included in the estimate of the wavelet local spectrum in
Equation 6.2. For fo = 5/2π, at 0.01 Hz nearly all the wavelet coefficients in that 128 s
time span are estimated from the same section of the time history, approximately 112 s
long (∆ti~56.27 s), and are thereby virtually non-unique. Thus their subsequent averaging
does little to reduce the variance, as conceptualized by Figure 6.6a. The figure illustrates
that at low frequencies there can be considerable overlap of the Morlet wavelet’s
Gaussian window within the analysis horizon T, yielding only three unique wavelet
windows, shown in white. Conversely, at higher frequencies, the same T affords five
unique wavelet windows. The ramifications parallel the estimation of power spectra from
Fourier-transformed blocks of a time history. Consider a signal of finite duration from
which five raw spectra can be generated only by heavily overlapping the blocks of the
time history being Fourier transformed. These five spectra are highly dependent and thus
only minimally reduce the variance when averaged. However, if the signal were long
enough to estimate five raw spectra from non-overlapping segments of the signal, their
averaged result would have far less variance, just as in the case of the higher frequencies
in Figure 6.4, whose wavelet coefficients are estimated using windows with temporal
duration of only a few seconds. The localized spectra in Equation 6.2 at these frequencies
include markedly more wavelet coefficients generated from independent segments of the
283
Low
Frequency
T
High
Frequency
t1
t3
t7
t8
t3
t7
t4
t5
t6
(b) VARIABLE INTEGRATION
t8
t2
(a)
t4
t5
t6
FIXED INTEGRATION
Low
Frequency
T(a)
High
Frequency
T(a)
t1
t2
FIGURE 6.6. Illustration of variable integration
window concept
time history. Now the same fo = 5/2π Morlet wavelet, at 0.5 Hz, produces a wavelet
coefficient from only 2.24 s of data (∆t~1.12 s), affording over fifty coefficients from
non-overlapping segments of the 128 s analysis window for averaging and sizeable
reductions in variance. This explains why lower frequencies in the coherence map seem
to be heavily plagued by spurious coherence, as emphasized when the coherence map is
plotted as semi-logarithmic in frequency in Figure 6.4.
The use of a fixed integration window (FIW) in Equation 6.2 actually provides
differential treatment to the high frequency components, in terms of the number of
uniquely estimated wavelet coefficients included in the averaging process. Due to the
multi-resolution character of the wavelet analysis, T in Equation 6.2 should be replaced
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by T(a), so that the integration in Equation 6.2 averages the same number of “ensembles”
over all frequencies, as also conceptualized in Figure 6.6b, where at both high and low
frequencies there are at least three unique wavelet windows, shown in white.
A scheme predicated upon variable integration windows (VIW) proceeds by
defining the window of integration for each frequency as an integer multiple (βT) of the
temporal resolution of the analyzing wavelet at that scale, given as
T (a ) ≥ 2 β T ∆t =
2βT a
2
=
2βT f o
f 2
(6.6)
for the Morlet wavelet. Thus βT would be chosen as a constant for the entire map,
dependent on the number of desired averages in the coherence measure, and T(a) would
vary, inversely proportional to the frequency being analyzed. The inequality in Equation
6.6 arises from the fact that the times at which the signal is sampled will not coincide
with the effective initiation and termination of an arbitrary dilated wavelet, such that T(a)
must be rounded to the nearest sampled point. This fact leads to the overlapping which
may occur at lower frequencies, as visualized in Figure 6.6. Note that Equation 6.6
insures that there is a minimum of βT independent time segments being windowed in the
estimation of wavelet coefficients, but there certainly may be additional overlapping
ensembles present, especially in the lower frequencies. As the number of independent
ensembles makes the most significant contributions to variance reduction, it allows the
simplest and most direct criteria for defining T(a).
285
In Figure 6.7, the benefits of variable integration are evaluated by comparing the
baseline case to three other cases: βT = 10, 20 and 50. Note that in the baseline case, the
fixed integration window yielded approximately βT = 50 in the high frequencies while
affording as little as βT = 1 in the low frequencies, accounting for the prevalence of
spurious coherence in this region. As shown in Figure 6.7, by averaging over a horizon
long enough to permit a sufficient number of wavelet coefficients to be estimated from
non-overlapping windowed sections of the time history, much of the low frequency
spurious coherence is minimized. The higher frequencies still appear to be plagued in
comparison to the baseline case, as expected, since the baseline essentially had βT =50 in
this region. Increasing the number of ensembles being averaged to βT =20 and 50 yields
further improvements in the higher frequencies, though the known pockets of coherence
are now beginning to bleed temporally, a consequence of increasing the time frame for
local averaging. This loss of temporal resolution in the coherence map is an unavoidable
consequence of increasing the number of ensembles in the averaging process. However,
the examples provided in Figure 6.7 illustrate that meaningless coherence can be
attributed to the effective number of ensembles in the averaging process in Equation 6.2
and justifies the use of a variable time window for this integration, as defined by
Equation 6.6.
Note that spectral averaging in general makes implicit assumptions of stationarity.
The stationary assumption is rarely justified in the wavelet analysis, as the suspicion of
evolutionary characteristics motivates the use of wavelet analysis in the first place. Still,
by viewing the problem as one with locally stationary blocks, this localized averaging
process can be reasonably justified. It is then left to the discretion of the user to determine
286
VIW: βT =10
Time [s]
Time [s]
Frequency [Hz]
Baseline: FIW
VIW: βT =50
VIW: βT =20
Frequency [Hz]
cW(a,t)
Time [s]
Time [s]
FIGURE 6.7. Examples of variance reduction by variable integration window
how transient the process is when specifying integration windows so that evolutionary
behavior is not masked. The averaging operation provides a more reliable description of
coherence as opposed to the raw coscalogram, providing a smoothed, yet biased estimate,
however it becomes evident that there is a delicate balance between a sufficient
integration window to minimize variance and one that is so long that it obscures relevant
time-varying information. In cases where highly evolutionary characteristics are
suspected, integration windows must be tightly localized, placing practical limits on the
use of localized integration to diminish spurious coherence, motivating more
sophisticated techniques to remove noise from the map, introduced below.
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6.4.2 Ridge Extraction by Hard Thresholding
As wavelet analysis is used commonly for the study of evolutionary behavior with
relatively short duration data, the possibility of significant amounts of averaging becomes
increasingly difficult if not impossible, particularly if the signal contains transient
information that would be completely obscured in the averaging process. An alternative
approach is to separate the signal from the noise surrounding it. In the case of analyzing
wavelets whose Fourier transforms are sharply focused near a fixed frequency value (e.g.
Morlet wavelet), the maxima of the resulting wavelet transform reflect the locations
where the energy of the signal concentrates, defining a curve in the time-frequency plane
termed the ridge – quite useful in situations where frequency-modulated signals are
imbedded in noise. Recall this concept was discussed more fully in Chapter 4 and has
been repeatedly exploited thus far in Chapter 5 and will continue to be revisited in future
chapters. Although noise is spread throughout the time-frequency plane, the contribution
of the signal is much greater than the noise in the vicinity of the ridges, which can be
extracted as discussed in Section 4.4.
However, in the coherence analysis presented here, the coherent pockets are
intermittent and not suitable for extraction techniques geared for smooth, continuous
ridges. Still, the theory of ridges implies that globally, as the truly coherent pockets will
show stronger coherence than the surrounding noise, leading to larger coefficients. As a
result, the truly coherent pockets may be separated by globally identifying the maximum
coherence (max[cw]) in the map and applying denoising schemes such as hard
thresholding. This process is defined by
288
⎧ 0
if
c W ,hard (a, t ) = ⎨ w
⎩c (a, t ) if
c w (a, t ) < λT max[c w ]⎫
⎬.
c w (a, t ) ≥ λT max[c w ]⎭
(6.7)
This process roughly approximates a ridge identification procedure, effectively
extracting the locations where the true coherence lies. Here λΤ is the assigned threshold
factor, taking on a value between 0 and 1 to define the percentage of the maximum
coherence deemed meaningful. The thresholding not only removes spurious coherence as
the result of variance, but also removes the effects of the Gaussian windowing operation
in the Morlet wavelet transform, which introduces a known enhancement of spectral
bandwidth discussed previously in Chapter 4. Though lesser values of coherence
surrounding a point in time and frequency are produced as a result of this window, the
highest coherences will still manifest along these ridge points, which carry all the
meaningful coherence information.
The thresholding operation in Equation 6.7 was applied to some of the cases
considered in the previous section, as the combination of ridge extraction by hard
thresholding and variable integration provides a simple means to extract meaningful
coherence from the wavelet coherence map. For βT =10, increasing the threshold factor to
0.75 approximately isolates both pockets of known coherence, as clearly shown when
comparing the results from Figure 6.7 to the filtered results in Figure 6.8. However, by
selecting too stringent a threshold (λT = 0.90, not shown), only a portion of the first
pocket of coherence is retained, while the second is completely lost. When the number of
ensembles is more sizeable, the threshold factor can be relaxed considerably, as βT = 50,
λT = 0.50 illustrates. Recall that this thresholding approach is merely another strategy to
289
βT =10, λT =0.75
βT =50, λT =0.50
βT =50, λT =0.75
Frequency [Hz]
βT =10, λT =0.50
Frequency [Hz]
Time [s]
Time [s]
FIGURE 6.8. Examples of coarse ridge extraction by thresholding
separate true coherence from noise. As the noise is primarily the byproduct of variance,
or a lack of averaging, in cases where variable integration has insured a large number of
ensembles in the average (e.g. βT =50), the noise has already been considerably alleviated
(see Fig. 6.7). In such cases, the noise is less dominant, taking on lower amplitudes in
comparison to the coherent ridges, thereby relaxing the necessary threshold value λT.
Note again, that the bleeding of temporal information, particularly for the first pocket of
known coherence, is an unavoidable consequence of increasing the number of ensembles
in the averaging process. Thus far, Sections 6.4.1 and 6.4.2 have discussed two
techniques for reducing the appearance of spurious wavelet coherence that, while simple,
are quite subjective. The following sections discuss more sophisticated techniques that
remove much of this subjectivity and provide an effective means to separate meaningful
pockets of coherence without requiring the use of variable integration windows.
290
6.4.3 Filtered Wavelet Coherence Map Based on White Noise
Though the coarse ridge extraction by thresholding is a simple means to identify
meaningful coherence, the insouciant use of hard thresholding based on global maxima
may obscure meaningful coherence that is weaker than the dominant coherent
components. Any coherence, real or noise-induced, falling below the threshold value is
neglected. Recognizing that the spurious coherence is the result of inherent randomness,
one alternative would be to conduct repeated Monte Carlo simulations of white or
colored random noise in order to determine the likely levels of variance in a given
wavelet spectral measure. By this approach, peaks in a wavelet scalogram, for example,
are deemed statistically significant if they surpass a given confidence level defined by the
random noise simulations. This type of approach was detailed in Torrence & Compo
(1998) for statistical significance of wavelet power spectra. J. Huang et al. (1998)
applied a similar technique to define a reference map for multi-resolution Fourier
cospectra. This approach is extended herein to wavelet coherence.
This process initiates with the generation of a reference coherence noise map
through repeated simulation of independent, zero mean, Gaussian white noise processes
at the same sampling rate and of the same duration as the original signals. The wavelet
coherence of the two simulated white noise signals is delineated by (ciwn (a,t )) and is
2
repeatedly calculated for N independently simulated white noise signal pairs, according to
Equations 6.1 and 6.2. These simulated coherence maps, which should represent spurious
coherence associated with random signals, are then averaged to produce a mean noise
reference map
291
(cmn (a, t ))2 =
1
N
∑ (c (a, t ))
N
i =1
wn
i
2
(6.8)
with standard deviation cst (a, t ) . The threshold value of a statistically meaningful
correlation can them be defined as the sum of this mean and the standard deviation
weighted by a factor g
cth (a, t ) = c mn (a, t ) + g (c st (a, t )) .
(6.9)
The noise factor g, which need not be an integer value, must then be selected
based on the desired likelihood of noise exceeding the threshold, providing a quantitative
measure of statistically meaningful correlation, where the distribution of the random
noise map is taken as Gaussian, consistent with the simulated noise. As a result, selecting
g = 1 implies a probability of suppressing 68.3% of the spurious coherence, g = 2 sets a
more stringent level at 95.4%, and g = 3 raises that level to 99.7%. With the reference
map cth(a,t) now in place, the actual wavelet coherence of x(t) and y(t) can be generated
by Equation 6.1 and then subjected to a thresholding scheme, yielding a filtered
coherence map according to
⎧⎪0
if
c (a, t ) = ⎨ W
⎪⎩c (a, t ) if
W
F
c W (a, t ) < cth (a, t )⎫⎪
⎬.
c W (a, t ) > cth (a, t ) ⎪⎭
(6.10)
The noise factor g in Equation 6.9 must be selected judiciously, as a choice that is too
large may negate statistically meaningful, albeit reduced, levels of correlation, as
observed in the thresholding procedure in the previous section. While the choice of g may
be rather arbitrarily defined, e.g. Hangan et al. (2001), a less subjective choice for g may
292
be determined based on the probability distribution of the noise coherence maps, as
recommended here.
To demonstrate the efficacy of this approach, N = 100 pairs of statistically
independent, zero mean, Gaussian white noise signals of the same length and sampling
rate of v1 and pr are simulated and the threshold of statistically meaningful correlation is
determined by Equation 6.9 for g = 1, 2 and 3. The unfiltered wavelet coherence map of
v1 and pr and the filtered maps for the varying levels of g are shown in Figure 6.9. By
progressively increasing the factor g, a more stringent thresholding condition is achieved.
The latter two figures clearly demonstrate the minimization of spurious coherence in the
low frequency range by virtue of this approach. Note that these results were achieved
without the use of any variable integration schemes but merely assuming ∆T = 64 s,
leading to the strong coherence in the low frequency range. By assuming a uniformly
distributed noise field and then performing the wavelet coherence operation, the spurious
coherence that results from implicit biases in the transform are recreated in the reference
thresholds and can be removed through the aforementioned thresholding operations
through minimal computational effort. For g = 3 the two known coherent pockets are
identified with some scattered, minor spurious coherent pockets remaining.
6.4.4 Filtered Wavelet Coherence Map Based on Signal Characteristics
The technique in the preceding section is more generalized assuming that a threshold
level representative of the spurious coherence can be generated using generic white noise
signals. However, this approach does not incorporate any specific information on the
spectral or probabilistic structure of the analyzed signals, perhaps leading to those
293
g = 1: 31.7% exceedence
Frequency [Hz]
Frequency [Hz]
unfiltered
g = 2: 4.6% exceedence
g = 3: 0.3% exceedence
Frequency [Hz]
Time [s]
Frequency [Hz]
Time [s]
Time [s]
Time [s]
FIGURE 6.9. Filtered wavelet coherence map between v1(t) and pr(t)
with g selected for varying noise exceedence levels, based on white
noise
residual trace coherent patches in Figure 6.9. As a result, a more sophisticated tool was
proposed by Dunyak et al. (1997) through a method to quantify the statistical relevance
of wavelet coefficients when detecting coherent wind gusts. These sustained gusts are
delineated from short incoherent bursts by establishing a reference distribution of wavelet
coefficients from simulated Gaussian signals with no sustained gusts. This notion can be
extended for the purposes of wavelet coherence analysis by employing a “smart”
thresholding scheme that exploits both the spectral and probabilistic information from the
294
signals being analyzed to generate a map describing the expected noise threshold (Gurley
et al., 2003).
In this case, the expected noise map associated with the wavelet coherence for a
given pair of signals, x(t) and y(t), is developed by first generating multiple simulations of
the second signal, denoted ys(t). These simulated signals are independent of each other
and x(t) and are statistically identical to the original signal y(t) in both the power spectral
density and probability density function, through the use of a recently developed nonGaussian simulation algorithm (Gurley & Kareem, 1997a). The wavelet coherence
between x(t) and each of the simulated versions of the second signal (ys(t)) is calculated
(
)
using Equation 6.1. The resulting wavelet coherence, denoted cinc (a,t ) , takes the place
2
of (ciwn (a,t )) in Equation 6.8. The mean noise reference map is then calculated by this
2
modified version of Equation 6.8 and the thresholding operations in Equations 6.9 and
6.10 are repeated using this “smarter” reference threshold incorporating the spectral and
probabilistic structure of the signals.
The non-Gaussian probability distribution of the random noise map in this case
must be approximated by considering higher-order statistics collected from the multiple
simulated noise maps, in addition to their mean and standard deviation. As the extreme
regions of the established distribution are used to determine the threshold, the probability
model used must reliably reflect the actual distribution of noise. An extreme value-type
distribution that does not explicitly include any information on the higher-order statistics
is disregarded in favor of more advanced four parameter models. A modified Hermite
polynomial-based model and a maximum entropy-based model are used in Gurley et al.
295
Frequency [Hz]
Frequency [Hz]
Unfiltered
25% exceedence
Time [s]
Time [s]
Frequency [Hz]
Frequency [Hz]
10% exceedence
1% exceedence
Time [s]
Time [s]
FIGURE 6.10. Filtered wavelet coherence map between v1(t) and pr(t) with g
selected for varying noise exceedence levels, based on signal characteristics
(2003). Both of these have been shown to be very effective in representing the tail region
of non-Gaussian processes (Gurley & Kareem, 1997b) and produced almost identical
results for the examples used in this chapter. The tail region of the resulting PDF
represents the probability of noise exceeding the selected threshold that demarcates
correlation in the wavelet coherence map.
To illustrate the proper selection of the factor g, in light of this modified
thresholding scheme, as well as the robustness of this approach, a filtered wavelet
coherence map is generated using N = 100 for the baseline analysis of the velocity and
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pressure signals, v1(t) and pr(t), discussed in Section 6.3.2. Note that this baseline case
also does not exploit the use of the variable integration windows (see Section 6.4.1) and
thus provides significant low frequency noise, just as the example in Section 6.4.3. As
Figure 6.10 illustrates, the performance of the unfiltered map is greatly enhanced as the
noise exceedence criterion is made more stringent. At the 1% exceedence level, both
regions of known coherence are completely isolated, even in the low frequency range,
indicating that the technique can remove spurious coherence that results from variance, in
the low frequency regime. Note that the trace pockets of spurious coherence that
remained in Figure 6.9 using a very strict thresholding condition of g = 3 were mitigated
fully by this “smarter” thresholding scheme thanks to the incorporation of the
probabilistic and spectral features of the signals themselves.
To further illustrate the application of the filtered wavelet coherence map,
measured full-scale incident wind velocity fluctuations and their corresponding pressure
variation over a building surface are analyzed. Note the strong low frequency correlation
evident in the time histories of velocity and pressure shown in Figure 6.11a,b. The
filtered wavelet coherence map is generated using a threshold reference map based on
500 simulated realizations of the data. Figure 6.11c-f shows the resulting filtered wavelet
coherence map at several levels of the noise factor g. Extraneous noise again is removed
as g increases, leaving a clearer portrait of the pockets of strong correlation. Though
relatively intensive, the application of these filtered wavelet coherence estimates to wind
data may be useful for identifying intermittent variations in the relationship between the
velocity and pressure introduced by a change in wind direction or due to the evolution of
a flow structure under the separation zone. Such maps can enhance the understanding of
297
Velocity
[mph]
(a)
Pressure
Coefficient
Time [s]
(b)
Time [s]
(d) 25%
Time [s]
Time [s]
Frequency [Hz]
Frequency [Hz]
(e) 10%
Frequency [Hz]
Frequency [Hz]
(c) Unfiltered
(f) 1%
Time [s]
Time [s]
FIGURE 6.11. (a) Measured wind velocity; (b) measured
wind pressure; (c) unfiltered wavelet coherence map;
filtered wavelet coherence map with g selected for varying
levels of noise exceedence: (d) 25% exceedence; (e) 10%
exceedence; (f) 1% exceedence
298
complex wind-structure interactions and open new avenues for data analysis, modeling,
and simulation.
6.5 Scale/Frequency-averaged Wavelet Coherence Map
Traditionally, coherence is displayed as a function of frequency only, averaged over the
entire time duration, as shown previously in Figure 6.3. As a result of the dual character
of wavelet transforms, the resulting coherence maps may be manipulated in order to view
coherence with respect to time. The wavelet coherence between two signals and those
between the first signal and simulated versions of the second signal are each averaged
over the scale component. In doing so, a display of the scale-averaged coherence, c (t ) ,
mean noise reference coherence, cmn (t ) , and threshold coherence, cth (t ) , between the two
signals can be generated with respect to time rather than frequency. Such an
interpretation for the simulated velocity and pressure, v1(t) and pr(t), is displayed in
Figure 6.12a. The intermittent correlated regions clearly stand out as those surpassing the
noise threshold, determined as a percent exceedence in the Hermite polynomial-based
probability distribution model derived from the first four moments of the noise coherence
maps as shown in Figure 6.12b.
6.6 Conclusions
In this chapter, wavelet decomposition was used to produce a time-frequency display of
the coherence between signals intermittently correlated. Unfortunately, raw spectral
estimates used in the definition of coherence were inherently laden with statistical noise.
299
Scale- averaged Coherence
Scale-averaged
Coherence
(a)
(b)
PDF
Time [s]
FIGURE 6.12. (a) Scale-averaged coherence along with mean coherence
and noise threshold; (b) Hermite polynomial-based PDF model of
reference noise maps
The classical approach for reduction of variance is to perform ensemble averaging by
using localized time integration. In this case, the introduction of a variable integration
window predicated on the multi-resolution character of wavelets highlighted that the lack
of ensemble averaging results in much of the observed spurious coherence. Insuring
sufficient ensembles in the average reduced the spurious coherence, though the loss of
temporal resolution was a limiting factor. As a result, a three-tiered thresholding
approach was introduced to isolate meaningful coherence. Hard thresholding, when
coupled with sufficient ensembles in the variable integration scheme, was shown to
enhance performance. However, to preserve evolutionary characteristics while removing
significant noise, more sophisticated approaches were required which do not involve
extensive averaging and VIW. The use of reference maps to filter the wavelet coherence
and distinguish meaningful coherence from bi-products of the transform was shown to be
quite effective in isolating known coherent pockets. This was first achieved by simply
using repeated simulations of white noise processes, which provided quite reasonable
300
results that could be further enhanced by the third tier: a “smart” thresholding simulation
scheme. In both approaches, the noise was filtered from the display map by comparison
with a threshold describing the likely noise level. In the latter approach, this threshold
was created by averaging a series of reference correlation maps between one signal and
uncorrelated simulations of the second signal. Examples demonstrated that this technique
was capable of identifying first-order correlation and effectively reducing the presence of
noise in the correlation displays for both simulated and measured data. Though relatively
intensive, these approaches facilitated the removal of significant levels of all of the
various contributing noise sources.
Unfortunately, even with such refinements, the wavelet fails to reveal the
instantaneous higher-order correlations that may exist in the transient spikes of
fluctuating pressures. On the other hand, while Fourier-based higher-order spectral
measures such as bicoherence can capture higher-order correlations, they have difficulty
detecting such intermittent nonlinear interactions. The same can be said for the case of
high amplitude, nonlinear extreme waves, whose first- and second-order components are
phase coupled over relatively short intervals (Powers et al., 1997). This motivated the
need for wavelet-based measures such as bicoherence to study nonstationary and
nonlinear characteristics of random waves and the resulting response of floating offshore
platforms, as well as prompting their consideration in Gurley et al. (2003) for higherorder intermittent correlation analysis of wind velocity and fluctuating pressures. In this
latter study, it is shown that this analysis framework can also be extended to a higherorder wavelet bicoherence measure to capture intermittent second-order correlation. In
total, when combined with the wavelet capabilities demonstrated in Chapter 5, the
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wavelet correlation detection analysis schemes presented here offer immediate
applications where the determination of intermittent correlation between linear and even
nonlinear processes is required, e.g. bluff body aerodynamics in turbulent flows, wavestructure interactions in nonlinear random seas, and the nonlinear and non-stationary
seismic response of structures. The next chapter will extend these wavelet frameworks to
the identification of frequency and damping in Civil Engineering structures.
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CHAPTER 7
WAVELET SYSTEM IDENTIFICATION III:
ANALOGS TO HILBERT SPECTRAL ANALYSIS
7.1 Introduction
Recently, the merits of the Hilbert transform with Empirical Mode Decomposition
(EMD), or Hilbert Spectral Analysis (HSA), have been explored for time-frequency
analysis by a number of studies. The use of this approach has been advocated widely by
illustrating its superior performance in comparison to the continuous wavelet transform
through a number of nonlinear and nonstationary examples. Although HSA itself
provides a new and useful tool for time-frequency analysis, at times, its comparisons with
wavelet transforms may be misleading and cast significant doubt on the wavelet’s ability
to provide satisfactory time-frequency analysis. This chapter shall revisit many of the
examples used to establish the Hilbert spectral analysis within the wavelet analysis
framework overviewed in Chapter 4 to offer a different perspective to these
commentaries, illustrating cases where the two approaches perform comparably and
highlighting situations where one is truly superior to the other. It is demonstrated that the
two approaches provide comparable evidence of nonlinear and nonstationary behavior for
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a number of classical examples, though this evidence is presented in distinctly different
manners with Hilbert spectral analysis relying solely on instantaneous frequency and the
wavelet using information from both this measure and the instantaneous bandwidth. The
intent of this chapter is not to advocate the use of one over the other, but rather to
objectively assess their efficacy and clarify some of the misconceptions surrounding
these applications.
7.2 Theory of Hilbert Spectral Analysis
As discussed in Chapter 3, the Hilbert transform cannot separate multiple frequency
components at any given point in time. Therefore, the application of this transform to
multi-component signals requires pre-processing of the data by bandpass filtering or
other appropriate methods to separate the various components (Lee & Park, 1994). Huang
et al. (1998) introduced the concept of Empirical Mode Decomposition to accomplish this
decomposition using empirical bases termed Intrinsic Mode Functions (IMFs), permitting
the application of the HT. These IMFs are defined so as to insure that they have wellbehaved Hilbert Transforms. These conditions are based on the fact that the instantaneous
frequency is accurately defined by restricting a function to be symmetric with respect to
the local zero mean level and by requiring that, within a given time series, the number of
zero crossings and extrema must be equal or at most differ by one. This latter condition is
established to insure that the IMF is monocomponent in nature. The term
monocomponent is somewhat ill defined, as discussed in Section 3.7.5; therefore, Huang
et al. (1998) had adopted a narrowband condition, based on the traditional characteristics
of a stationary, narrowband Gaussian process.
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While the full details of EMD can be found in Huang et al. (1998), the procedure
can be summarized as follows, retaining here the notation used in that study. The process
initiates with identification of the local extrema. These extrema are then used in
conjunction with a cubic spline fit to construct an upper and lower envelope for the
signal. The average of the upper and lower envelopes, as a function of time, is defined as
the envelope mean, which may not correspond to the local signal mean. The difference
between the signal and envelope mean is defined as the first component, h1.
To eliminate riding waves and make the wave profiles more symmetric, this first
component h1 is treated as the data, and the process of fitting extrema and determining
their mean is repeated in the so-called sifting process such that
h1k = h1( k −1) − m1k
.
(7.1)
Thus, in the kth iteration, the previous component’s extrema are fit and the mean is again
removed. This process of sifting is repeated until the two conditions placed on a proper
IMF are satisfied. Then the resulting sifted data is treated as the first IMF component,
c1=h1k. This component will contain the finest scale. This finest scale represented by the
first IMF is then removed from the data, leaving the residue,
r1=x(t)-c1
(7.2)
and the residue r1 is now treated as the data and the entire sifting process is repeated to
obtain the second IMF, c2, and so on to obtain all remaining IMFs such that
rn = rn-1 – cn .
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(7.3)
The process will terminate when rn or cn become very small, or rn is essentially
monotonic. The subsequent IMFs obtained in this manner will progressively capture the
coarser scales remaining in the signal. The signal can then be reconstructed by the
superposition of these N IMFs according to
N
x(t ) = ∑ c i + rN
i =1
(7.4)
It should be noted that the determination of appropriate IMFs is predicated on the
restrictions levied by the Hilbert Transform to provide an accurate instantaneous
frequency, requiring the IMFs to be symmetric with respect to the local zero mean and
have the same number of zero crossings and extrema. If the IMFs generated in this
process meet these conditions, then the monotonic signal conditions surrounding the HT
should be satisfied and the HT can be applied to each IMF to obtain an estimate of the IF
with time. Constructed in this fashion, the IMFs are entirely empirical, and a given IMF
may even contain two distinct frequency components, though at each point in time, the
IMF is entirely monocomponent. Thus, the decomposition is different from Fourier
transforms, where each component is related to a given frequency for its entire duration.
In this decomposition, the data are expanded in a basis derived from the data itself.
As Huang et al. (1998) reiterates, an IMF on its own has little meaning and the
true physical meaning can only be garnered from the complete Hilbert spectrum. This
Hilbert spectrum (HS) is generated by converting each IMF into an analytic signal by
Equation 3.23. The resulting amplitude of the analytic signal for each IMF is plotted as a
dual function of instantaneous frequency, as determined by Equation 3.27, and time. The
result is often viewed in two dimensions by studying the contours of the HS in the time306
frequency plane. This total process employing EMD and the HT is termed the Hilbert
Spectral Analysis. The fact that the performance of wavelet transforms are heavily
dependent upon the parent wavelet chosen was a motivating factor for developing the
HSA, as the IMFs serving as basis functions are derived from the data itself and thus
maintains physical resemblance.
7.3 Preliminaries
It should be noted that the ability of this technique to characterize the instantaneous
frequencies within the data rests upon two critical elements: first upon the ability of EMD
to separate the various components within the signal, and secondly, upon the ability of
the Hilbert Transform to truly yield the quadrature component of the signal, dictating the
accuracy with which the instantaneous frequency is identified. As will be discussed
throughout this chapter, limitations in either one of these areas can affect the performance
of Hilbert Spectral Analysis. Before discussing these and other issues by example, a few
additional preliminary discussion points must be established.
7.3.1
Resolutions of Hilbert Spectral Analysis
The accuracy of IF estimation is dependent upon the resolutions of the time-frequency
transform. The resolutions associated with the Morlet wavelet transform were discussed
in considerable detail in Chapter 4. On the other hand, the resolutions of the Hilbert
Transform are difficult to clearly quantify. The transform’s temporal localization is
verifiable, as the signal is being convolved with 1/t in Equation 3.24. As a consequence
of the Heisenberg Uncertainty Principle, the good localization in the time domain implies
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poor frequency resolution, necessitating the incorporation of bandpass filtering, or in this
case EMD, to insure modal separation. As a result, the inherent accuracy of these preprocessing tools indirectly dictates the frequency resolution of HSA. Therefore, the
frequency resolution of HSA is based upon EMD’s frequency resolution, essentially the
narrowband condition on the maxima and zero crossings. The implications of this will be
discussed in subsequent examples.
7.3.2
End Effects in Hilbert Spectral Analysis
Just as with wavelet transforms, the implementation of numerical schemes using Fourier
transforms can introduce end effects. It was shown in Chapter 4 that these end effects are
attributed to this but more importantly to an additional source in wavelet transforms.
Huang et al. (1998) had also observed end effects, attributed to the spline fitting in the
EMD procedure and due to the fact that the Hilbert Transform was implemented via
Fourier Transforms. The authors remedied this using characteristic waves to extend the
data and reduce discontinuities at the ends. This can be likened to the reflective padding
scheme presented for wavelets in Chapter 4.
7.3.3
Introduction of Instantaneous Bandwidth
While the notion of instantaneous frequency has been generally accepted in the literature
as the average of the frequencies at each point in time (Priestley, 1988), the discussion
has not fully extended to consider the instantaneous bandwidth or spread of frequencies
contributing to this mean. In recent years, a few definitions of instantaneous bandwidth
have surfaced within the literature, though these discussions have not fully demonstrated
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the utility of this measure. However as the examples in this chapter will later show, in the
case of wavelet analysis in nonlinear systems, the spread about the instantaneous
frequency is equally as important as the mean frequency itself.
As discussed in Cohen & Lee (1990), the squared bandwidth can be classically
defined as
∞
B2 =
∫ (ω − ω ) S (ω )
2
2
dω
(7.5)
−∞
where S(ω) is the Fourier transform of some arbitrary signal s(t) and <ω> is the mean
frequency defined by
∞
ω = ∫ ω S (ω ) dω
2
−∞
.
(7.6)
As discussed in Jones & Boashash (1990), these spectral measures in Equations 7.5 and
7.6 can similarly be normalized by the area under the power spectrum. Regardless, this
bandwidth can then be related to the amplitude and phase of the signal by
∞
B =
2
∫ A′
−∞
∞
2
(t )dt +
∫ (φ ′(t ) − ω )
2
−∞
A 2 (t )dt
(7.7)
where the first term is independent of phase and the second defines the deviation of the
instantaneous frequency from the mean frequency of the signal. What this relation
implies is that contributions to the signal bandwidth arise from two sources, the
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amplitude variations (first term) and frequency variations (second term), i.e. from AM
and FM sources.
Replacing S(ω) with a time-frequency energy distribution (e.g. Wavelet or Short
Time Fourier Transform), changes Equation 7.6 into an expression for instantaneous
frequency or mean frequency of an instantaneous power spectrum. Similarly Equation 7.7
becomes a measure of instantaneous bandwidth or the deviation from the instantaneous
spectral mean.
Cohen & Lee (1990) later show that an instantaneous bandwidth measure can be
equivalently obtained from the analytic signal itself. Consider again the analytic signal
introduced in Equation 3.25. The amplitude term AZ(t) can be viewed as an envelope
function, and provided that the this envelope varies more slowly than the phase, as
discussed previously in the theory of asymptotic signals in Section 3.7.2, the
instantaneous frequency can be identified from the derivative of the phase. According to
Cohen & Lee (1990), the instantaneous bandwidth can then be obtained from the
derivative of the analytic signal’s envelope:
2
⎛ A′ (t ) ⎞
Bi (t ) = ⎜⎜ Z ⎟⎟
⎝ AZ (t ) ⎠ .
(7.8)
Based on this definition, the concepts of instantaneous frequency and bandwidth make
sense only for signals that are locally narrowband (Ristic & Boashash, 1996). However,
Boashash (1992a) has been critical of this definition, as it focuses entirely on the
amplitude modulations and implies that if no amplitude modulation is present, i.e. Az(t) is
310
constant, then the signal would have zero bandwidth and only one frequency present.
These intuitive difficulties have presented concerns for this definition, though in the case
of amplitude modulated signals, it has been shown by Cohen & Lee (1989) that this
definition produces the same value of instantaneous frequency as the classical definition.
Despite this attractive theoretical framework, the resulting instantaneous
bandwidth measures are not always physically meaningful. For example, Jones &
Boashash (1990) concluded that the Wigner-Ville and Choi-Williams distributions could
produce negative spectral bandwidths due to the inherent negativity in the distributions
themselves. As a result, only a measure of spread derived from a positive definite
distribution such as the Short Time Fourier Transforms are physically useful, as
demonstrated by Barnes (1992) in the analysis of seismic reflection data and the
aforementioned examples by Cohen & Lee (1989). Fortunately, the wavelet similarly
provides the opportunity for meaningful analysis of instantaneous bandwidths.
The calculation of derivative in the instantaneous bandwidth of Equation 7.8 can
be difficult, prompting authors to use other approaches, e.g. maximum likelihood
estimators (Ristic & Boashash, 1996). Similarly, approaching the problem as an integral
operation on an instantaneous spectrum, as suggested by Equation 7.5, is complicated for
multi-component signals, as it can be difficult to isolate the spectral band associated with
each component in order to identify its instantaneous bandwidth. In such cases, limits of
integration must be imposed, a task which is somewhat ambiguous and arbitrary. As a
result, it is suggested herein to make use of a different estimator that is localized in the
vicinity of the spectral peak, making it easy to apply in multi-mode scenarios and
311
circumventing the aforementioned concerns. The half-power bandwidth measure first
introduced and discussed in Section 4.2.2.2 can provide such an estimator in a simplified
and straightforward manner. Though arbitrary in its definition, this bandwidth measure
allows the tracking of variations of the instantaneous bandwidth in time to uncover the
presence of nonlinearity in the signal, within a computationally efficient framework. It is
this relative bandwidth variation that is of most utility in the subsequent discussions.
Finally, it should be noted that the spectral bandwidth is often exaggerated by the
window effects introduced by local transforms. This detail must be kept in mind when
analyzing instantaneous spectral bandwidths. Cohen & Lee (1989, 1990) investigated the
relationship between signal bandwidth and window bandwidth for Short-Time Fourier
Transforms. Though the exact relations can be found in the cited reference, the study
overall conveys the fact that the bandwidth of the spectrogram (BSP) includes both the
signal’s true bandwidth Bs and the Bw bandwidth associated with the window of the
transform:
2
B SP
= B s2 + B w2
.
(7.9)
Cohen (1999) has also investigated the moments of the wavelet transform. Although he
noted the more complicated entanglement that occurs as the result of scaling, the first and
second moments contain the contributions of both the window and the signal bandwidth,
as noted previously for the Short Time Fourier Transform. Despite this fact, the halfpower bandwidth measure presented earlier in Chapter 4 can be used to identify relative
changes in spectral bandwidth, as the window contributions remain constant throughout
the analysis. These relative fluctuations in bandwidth represent variations in the range of
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frequencies present locally in time, vital information for nonlinear systems, when viewed
in conjunction with the dominant frequency in that locality, provided by the traditional
instantaneous frequency.
7.4 Proper Conditions for Comparison of Hilbert Spectral Analysis and
Wavelet Transforms
A critical reason for advocating the use of HSA over a WT analysis has been its ability to
provide a clearly defined time-frequency skeleton plot. In contrast, the WT is shown in
Huang et al. (1998) to provide a somewhat smeared and less precise representation as
frequency ranges are plotted in a color contour map. However, by examining closer the
theory behind both approaches, one can identify the rationale behind this discrepancy in
performance. Recall that the wavelet transform is inherently a two variable transform,
being a function of scale (frequency) and time. Thus, a wavelet analysis directly provides
an energy density that is simultaneously a function of both frequency (scale) and time. In
particular, as the Morlet wavelet considers windowed sections of the signal in its
analysis, these windows modify the bandwidth of the signal, as discussed in the previous
section. This characteristic helps to explain the “smeared” appearance of a continuous
wavelet scalogram as noted in Huang et al. (1998), though the extent of this leakage can
be minimized by adjusting the Morlet wavelet’s central frequency.
Like the WT’s scalogram, Hilbert spectra also yield the time-frequency energy
density; however, in this case the transformation is not inherently a function of both
frequency and time. Instead, the use of the HT allows the complex analytic signal to be
formed, which is not a function of frequency but merely a function of time. It is only in
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the subsequent calculation of the instantaneous frequency from the phase of this analytic
signal that a frequency variable can be obtained for the HSA. By calculating the
frequency from the phase relationship, the Hilbert spectrum provides a single frequency
coordinate for each component in the signal. This is an important distinction that helps to
further explain the reason for the poor performance of the WT in its initial comparisons
to the HSA in Huang et al. (1998).
Since the WT provides a complex signal directly, as a function of both frequency
and time, it too can be used to determine the instantaneous frequency of the signal. In a
similar manner, the scalogram values at these instantaneous frequencies may be plotted at
each point in time to yield the wavelet ridge (Staszewski, 1997; 1998). Such
representations, defined previously in Chapter 4, are termed wavelet instantaneous
frequency spectra in this study, to simply reinforce its parallels with the Hilbert spectra.
In this way, the performance of the HSA and WT can be compared via equivalent
standards by examining the ability of each to identify the evolution of the instantaneous
frequency with time. It should be reiterated that the HT would not be able to provide an
estimate of frequency without invoking the calculation of instantaneous frequency, just as
the WT will never be able to provide a point estimate of frequency without also
calculating an estimate of IF. As previous comparisons in Huang et al. (1998; 1999) did
not consider this, the WIFS is used throughout this chapter to revisit these continuous
wavelet comparisons to the HSA.
314
7.5 Examples
The following sections shall revisit a number of examples presented originally in Huang
et al. (1998), examining the HSA results against revised Morlet wavelet analyses
performed within the framework offered by this study. The current continuous wavelet
results simulated by Equation 3.9 feature enhanced resolution by adjusting the central
wavelet frequency, include the proper discretization and appropriate padding for end
effects, discussed in Chapter 4, and are presented in a form analogous to the HS via the
WIFS. In these comparisons, Hilbert spectra adapted directly from Huang et al. (1998),
which provided the first meaningful comparison between HSA and WT, will be
considered in order to give complete justice to HSA. Any differentiation of phase data to
estimate the instantaneous frequency is accomplished by determining the slope of a
piecewise linear fit to the phase data by a least squares approach. For comparison, the
original continuous wavelet analyses conducted by Huang et al. (1998) are also provided.
Note that the quality of these analyses depend on the discretization of the time-frequency
domain as well as the central frequency fo of the Morlet wavelet applied. Specific details
on these parameters used in Huang et al. (1998) were not provided.
7.5.1 Example 1: Localized Sine Wave
The first example presented is a single cycle of a 1 Hz sine wave. Figure 7.1a,b displays
the signal and the scalogram generated using the Morlet wavelet. What should be noted
in this case is that an analysis of such transient phenomena requires a wavelet with a
localized temporal resolution, adjusting fo to 1 Hz. The dark red patch in the center of the
wavelet map has pinpointed the time location and frequency content of the signal. The
315
(a)
(b)
t [s]
f [Hz]
f [Hz]
f [Hz]
f [Hz]
x(t)
(d
(c)
(e)
t [s]
t [s]
FIGURE 7.1. Example 1: (a) isolated cycle of 1 Hz sine wave; (b) wavelet
scalogram; (c) WIFS (red dotted) with frequency of sine wave (blue dashed);
(d) Huang et al.’s (1998) Morlet wavelet result; (e) Huang et al.’s (1998)
Hilbert spectrum
“bleeding” of the wavelet scalogram is the result of inherent window effects in the time
and frequency domains; however, these Gaussian windows will produce maximum
wavelet coefficients at the time and frequency where the signal’s energy content is
concentrated, pinpointing the ridge. Using a global search for maxima of the wavelet
modulus, the actual value of instantaneous frequency identified by the wavelet is shown
by the WIFS in Figure 7.1c. By this approach, the wavelet will precisely identify the
instantaneous frequency of the signal in the range of 4.1-4.2 seconds and 4.8-4.9 seconds
as 0.98 Hz. This identification becomes even more exact at the heart of the transient,
between 4.3 and 4.7 seconds with the identified value of 1.08 Hz. The continuous wavelet
transform was previously criticized by Huang et al. (1998) for using spurious harmonics
to isolate the transient (Fig. 7.1d). However, the wavelet analysis in Figure 7.1b does not
316
rely on higher harmonics, as the maxima of the transform, colorized in red, are restricted
to the 1 Hz range. In fact, the localization in time and frequency of the WIFS is
comparable to that generated by the HSA (shown in Figure 7.1e) and may even surpass
its performance at the initiation and termination of the single cycle of oscillation.
7.5.2 Example 2: Sine Wave with Frequency Discontinuity
Another example of a sudden change of a signal’s frequency is given in Huang et al.
(1998) by a 0.03 Hz sine wave that suddenly shifts to a 0.015 Hz frequency of oscillation.
The occurrence of this shift is highly localized in time, occurring suddenly at the 500th
second, as shown in Figure 7.2a. In the continuous wavelet analysis by Huang et al.
(1998), the smearing of information in both frequency and time is highly undesirable
(Fig. 7.2e) and pales to the pinpoint precision of the HSA (Fig. 7.2d). Revisiting the
continuous wavelet example again using a Morlet wavelet with better frequency
resolution properties (fo=5 Hz), two distinct frequency bands are identified in Figure 7.2b,
although the transition between the two is masked by an apparent smearing of the wavelet
coefficients due to window effects. However, upon examining the WIFS, a plot of the
identified wavelet instantaneous frequency, analogous to the quantity portrayed in the
Hilbert spectrum, is shown in Figure 7.2c and illustrates the precision with which the
continuous wavelet can identify both the frequency and time characteristics of this signal.
7.5.3 Example 3: Quadratic Chirp
To examine if the temporal smearing of the scalogram is problematic for a case where
nonlinear frequency changes occur continuously throughout the signal, the example of a
317
(a)
(b
(e)
f [Hz]
f [Hz]
f [Hz]
x(t)
f [Hz]
(d)
(c)
t [s]
t [s]
FIGURE 7.2. Example 2: (a) cosine wave with frequency halved midway through
signal; (b) wavelet scalogram; (c) WIFS; (d) Huang et al.’s (1998) Hilbert
spectrum; (e) Huang et al.’s (1998) Morlet wavelet result
quadratic chirp is considered. The chirp (shown in Figure 7.3a) quadratically varies from
0.2 Hz at 0 seconds to 0.17 Hz at 120 seconds. Hilbert and continuous wavelet (fo=3 Hz)
analyses were conducted, with the scalogram shown in Figure 7.3b and the identified
instantaneous frequencies for the Hilbert transform (red) and continuous wavelet
transform (blue) shown in Figure 7.3c. Note that the analysis of this chirp signal does not
require pre-processing by EMD, as it possesses a unique frequency component at each
instant in time. Both capture the quadratic variation in frequency, despite some end
effects problems. The padding operation discussed previously is performed on the
quadratic chirp signal and the continuous wavelet and Hilbert transforms are again
determined. The result in shown in Figure 7.3d-e and demonstrates that the padding
operation, not only enhances the performance of the continuous wavelet transform, but
improves the Hilbert result as well – diminishing the large swings at the ends. Note
318
f [Hz]
x(t)
(a)
(b)
f [Hz]
(c)
f [Hz]
(d)
f [Hz]
(e)
t [s]
FIGURE 7.3. Example 3: (a) quadratic chirp; (b)
wavelet scalogram; (c) identified instantaneous
frequency by WT (in blue) and HT (in red); (d)
wavelet scalogram using padded signal and finer
discretization of scales; (e) identified instantaneous
frequency by WT (in blue) and HT (in red) and
actual quadratic frequency law (green) using padded
signal and finer discretization of scales
319
however that the two transforms treat the signal in different ways. The HT manifests
oscillations of the instantaneous frequency in addition to the global quadratic decay. The
WT does not detect this oscillatory behavior and instead provides equivalently piecewise
fit of the IF law to capture the overall quadratic nonlinearity. Note that in Figure 7.3c, the
nature of this piecewise fit was rather coarse; however by choosing a finer discretization
of scales in the wavelet analysis in Figure 7.3e, the identified IF law is far smoother. This
piecewise fit arises from the fact that the wavelet fits small waves or “wavelets” to the
signal at each point in time – as expected, a locally linear approximation to the quadratic.
Note that in Figure 7.3e, the actual quadratic frequency relation used to simulate the chirp
is plotted in green to demonstrate the accuracy of HT and WT in identifying the
frequency variations. The WT result in blue nearly overlays identically with the
theoretical result in green. The HT result in red is slightly less accurate and manifests an
oscillatory behavior that is not present in the actual quadratic IF law. Thus far, Examples
2 and 3 respectively illustrate the equal ability of both techniques to identify changes in
frequency occurring over a range of frequencies and at an isolated instant in time.
7.5.4 Example 4: Stokes Wave
The true significance of the Morlet wavelet’s central frequency can be discerned in the
example of the idealized Stokes wave in deep water. The use of a perturbation analysis to
solve this system illustrates the common practice of representing nonlinear phenomena as
a summation of harmonic components, with the second-order approximation to the
Stokes wave profile given by
320
X (t ) =
1 2
1
a k + a cos(ωt ) + a 2 k cos(2ωt )
2
2
(7.10)
in which a is the amplitude and k is the wave number. By choosing a=1, k=0.2 and
ω=2π/32, as discussed in Huang et al. (1998), Figure 7.4a is generated. By inspecting the
instantaneous wavelet modulus taken at an instant in time, as demonstrated in Figure
7.4b, a component at 0.0320 Hz is evident, representing the 0.0313 Hz harmonic. The
presence of a higher mode at 0.0622 Hz, though evident only in the wavelet modulus in
Figure 7.4b due to its small magnitude relative to the dominant ridge of the scalogram in
Figure 7.4c, confirms the presence of the 0.0625 Hz component. Figure 7.4c displays the
resulting Morlet wavelet scalogram for fo=5 Hz, whose color scale does not clearly reflect
the second mode, though the WIFS in Figure 7.4d detects the two frequencies identified
from the local maxima shown in Figure 7.4b. The Morlet wavelet with 5 Hz central
frequency, though having a refined frequency resolution, will have a poor temporal
resolution of over 100 seconds. Thus it behaves much like a Fourier harmonic analysis
and is well suited to capture the dual harmonics used in Equation 7.10 to simulate the
second-order approximation to the Stokes wave. It unfortunately has no ability to capture
the local nonlinearities of the wave profile. Note that the degree of smearing has been
minimized in the current continuous wavelet analysis in Figure 7.4c in comparison to the
previous findings in Figure 7.4e. This perspective offered by the wavelet analysis reflects
the philosophy used to simulate the nonlinear system – represent the nonlinearity by a
sum of harmonics.
In contrast, Figure 7.4f displays the HSA result, which oscillates about 0.0313 Hz
and shows no evidence of the second mode. To justify this result, Huang et al. (1998)
321
|W(a,t=250s)
|
(c)
(d)
(e)
(f)
x(t)
(a)
(b)
f [Hz]
f [Hz]
f [Hz]
t [s]
t [s]
FIGURE 7.4. Example 4: (a) Second order approximation to the Stokes wave; (b)
wavelet modulus at t=250 s; (c) wavelet scalogram; (d) WIFS; (e) Huang et al.’s
(1998) Morlet wavelet result; (f) Huang et al.’s (1998) Hilbert spectrum
explained that Stokes waves could similarly be modeled as a cosine with modulated
frequency
X (t ) = cos(ωt + ε sin(ωt ))
,
(7.11)
which is much like the frequency modulated signal in communication theory, with ω
serving as the carrier frequency and the second frequency term being the transmitted
signal. As argued by Huang et al. (1998), if there are any frequency changes within the
wave’s oscillation, then its profile can no longer be a simple sinusoid. A visual inspection
of the simulated Stokes wave in Figure 7.4a demonstrates that there is such a subtle
departure from the simple sinusoidal shape. This deviation from a simple sinusoid
322
observed in Stokian waves is the result of intrawave frequency modulation, represented
by the second frequency term in Equation 7.11.
An enhanced temporal wavelet analysis echoes this behavior. Inspecting the time
series, peaks have a frequency of around 0.03 Hz or a period of about 32 s. The resolution
of the wavelet in both time and frequency must be sufficient to capture this. From the
relations in Equation 4.5 and Equation 4.8, a central frequency of 0.5 to 1 Hz will be
required to balance the time and frequency resolution capabilities. Thus the wavelet
analysis of the Stokian wave simulated by Equation 7.10 (shown again in Figure 7.5a) is
repeated with fo=0.5 Hz. In this analysis, the scalogram (in Figure 7.5b) still concentrates
near 0.03 Hz, however there is no evidence of a higher harmonic. Instead there is an
oscillatory variation toward the high frequency range, evidenced by the light blue hues.
Inspection of the WIFS (Figure 7.5c) still does not confirm this, as it manifests a single
frequency component at 0.0303 Hz that does not vary with time. Looking at the
instantaneous frequency estimated from the phase of the wavelet transform, a nearly
constant frequency value is again identified, though more precise, at 0.0312 Hz. Still, the
scalogram behavior indicates that the wavelet is detecting some variation of evolutionary
energy content. As discussed previously in Section 7.3.3, the spectral characteristics of a
system are not merely defined by the instantaneous frequency. As a result, the bandwidth
of each instantaneous spectra produced from the wavelet analysis is provided in Figure
7.5d and demonstrates that this value oscillates about a mean value with a period of
approximately 32 seconds in the same manner as the HS in Figure 7.4f.
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x [ft]
(a)
f [Hz]
(b)
Β [Hz]
f [Hz]
(c)
(d)
t[s]
FIGURE 7.5. Example 4 via refined wavelet analysis: (a)
second order approximation to the Stokes wave; (b)
wavelet scalogram; (c) WIFS; (d) wavelet instantaneous
half-power bandwidth
Application of a similar wavelet analysis on measured surface elevation of wave
tank data verifies these characteristics of Stokian waves. Figure 7.6a displays wave data
generated mechanically by a 1 Hz sinusoidal excitation with +/- 9 mm amplitude. Note
that the time series manifests narrowed peaks and widened toughs, highlighting a subtle
deviation from a simple sinusoidal shape. By inspection of the time series, the peaks are
separated by approximately 1 s, necessitating a temporally refined wavelet capable of
determining variations occurring over less than one second and frequency sensitivity a
324
x [ft]
f [Hz]
f [Hz]
(a)
(b)
(c)
f [Hz]
(d)
f [Hz]
Β [Hz]
(e)
(f)
t[s]
FIGURE 7.6. (a) 30 seconds wave tank surface elevation; (b)
scalogram of wave tank data; (c) WIFS; (d) instantaneous
frequency from wavelet phase; (e) instantaneous half-power
bandwidth estimate; (f) instantaneous frequency from Hilbert
transform phase
325
fraction of 1 Hz. With fo=0.5 Hz, a temporal resolution of 0.35 s is possible, with a 0.22
Hz frequency resolution. The scalogram for this analysis, shown in Figure 7.6b, again
concentrates near 1 Hz with its warm hues, but the lighter shades of blue fluctuating in
the higher frequency scales again indicate the presence of time-varying frequency
content. The WIFS in Figure 7.6c, as observed in the previous example of Stokian waves,
remains constant at 0.94 Hz, giving an averaged interpretation of the instantaneous
frequency as a result of the wavelet approximation. Zooming in on the more precise
phase-based instantaneous frequency estimate, in Figure 7.6d, the minor modulations
reveal time-variance in the local mean frequency. Further fluctuations about this mean
frequency are then identified in the instantaneous bandwidth in Figure 7.6e, which
provides a rich display of nonlinear characteristics beyond that of the numerically
simulated Stokes waves in Figure 7.5. The bandwidth in this case oscillates again at the
frequency identified in the WIFS, however, the periodic modulations of the bandwidth
indicate a regular variation of frequencies concomitant at each instant in the signal. A
Hilbert Spectral Analysis by Huang et al. (1998) of measured wave data affirmed similar
phenomena, albeit displayed solely in the instantaneous frequency. The direct application
of the Hilbert transform in Figure 7.6f can affirm the variations of the frequencies present
in the system, though this perspective is far noisier without the filtering afforded by
EMD.
This example illustrates two important facts: first, that a wavelet analysis with
poor temporal resolution inherently treats the signal in the same manner as Fourier
analysis, while a wavelet analysis with refined temporal resolution is capable of detecting
nonlinear wave characteristics. It is the resolutions tied to a specific analysis that
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ultimately dictate whether the nonlinear system will be viewed in terms of harmonics or
intrawave modulated waves, i.e. waves deviating from a simple sinusoid with subcyclic
oscillations. While the resolutions of HSA, though temporally precise, remain fixed, the
flexibility in wavelet transforms allowed both of these perspectives to be realized.
Secondly, but perhaps more importantly, the wavelet does not necessarily manifest these
indicators in the instantaneous frequency, but instead in the instantaneous bandwidth.
Recall again that the wavelet fits small waves or “wavelets” to the signal at each point in
time. In the case of the Morlet wavelet, these localized waves are sinusoidal in nature.
The IF identified in Figure 7.5c is then the frequency of the best fit widowed sinusoid.
However, as the Stokian wave profile subtlety deviates from the simple sinusoid, it is not
unreasonable to expect that additional neighboring frequencies are required to capture
these deviations. The involvement of such adjacent frequencies is represented by the
bandwidth measure. Thus in the truest sense indicated by Priestly (1988), the wavelet IF
is the mean frequency and the bandwidth reflects the deviation of these frequencies from
this mean as they evolve in time.
7.5.5 Example 5: Linear Sum of Two Closely-Spaced Cosines
As the previous example demonstrated, the temporal resolution of HSA is fixed, yet
precise. The time resolution is a consequence of the Hilbert transform’s convolution in
Equation 3.24 involving a highly localized function. The following example will
demonstrate a situation in which the sacrificed frequency resolution in HSA can be
problematic. A pair of closely spaced cosine waves was generated by
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(c)
(d)
(e)
f [Hz]
f [Hz]
x(t)
f [Hz]
(b)
f [Hz]
(a)
t [s]
t [s]
FIGURE 7.7. Example 5: (a) linear combination of two closely-spaced cosine
waves; (b) wavelet scalogram; (c) wavelet scalogram (zoom); (d) WIFS; (e) Huang
et al.’s (1998) Morlet wavelet analysis with Hilbert spectrum contours
superimposed
⎛ 2 ⎞
⎛ 2 ⎞
x(t ) = cos⎜ πt ⎟ + cos⎜ πt ⎟
⎝ 30 ⎠
⎝ 34 ⎠
(7.12)
The frequencies of these two harmonics are approximately 0.0294 and 0.0333 Hz. Figure
7.7a shows the signal with characteristic beat phenomena. In order to separate the two
components, a given analysis technique must have a refined frequency resolution,
consistent with the findings of Delprat et al. (1992). In Huang et al. (1998), it was shown
that neither a continuous wavelet analysis nor HSA could identify two distinct harmonic
components, as shown in Figure 7.7e. Revisiting this problem using fo=5 Hz for the
Morlet wavelet, presumably different from Huang et al.’s (1998) analysis, the wavelet
scalogram (Fig. 7.7b) was generated. As the zoom in Figure 7.7c illustrates, two distinct
bands of energy are revealed, shown in red and light blue. Searching the localized
maxima, two instantaneous frequency components can be identified at 0.0303 Hz and
0.0340 Hz, within 3% of the actual signal frequencies, as shown by the WIFS in Figure
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7.7d. Figure 7.7e again displays the wavelet scalogram obtained by Huang et al. (1998)
with the Hilbert spectrum superimposed as contours. Though the signal is the linear
combination of two distinct harmonics, according to Huang et al. (1998), neither
approach accurately captures this. The wavelet scalogram indicates a high concentration
of energy at approximately 0.03 Hz, but does not distinguish between the two
components. However, the continuous wavelet’s inability to separate the harmonics in
Huang et al. (1998) should not be interpreted as a failure of the continuous wavelet in
theory, but rather a byproduct of the selection of an insufficient central frequency fo in the
analysis.
Likewise, the Hilbert spectrum localizes in the same vicinity but also shows some
spurious oscillatory behavior in the instantaneous frequency between 0.025 Hz and 0.035
Hz, treating the beat phenomena as a frequency modulated wave. As shown in the
previous example, the presence of multiple components in an IMF will result in nonlinear
phase terms once the HT is applied. In such cases, the HT treats the closely spaced
harmonics by some non-physical modulated wave. The misrepresentation in this example
may be a direct consequence of EMD. In this case of two closely spaced modes, the EMD
required a very stringent condition of 3000 siftings to obtain only eight IMF components,
which still could not represent the true signal (Huang et al., 1998). The inability to
distinguish between two distinct components may be traced to the definitions of the IMFs
to satisfy the restrictions of the Hilbert transform. As discussed previously, Huang et al.
(1998) used a more relaxed narrowband condition, placing conditions on the number of
zero crossings and maxima. In doing so, the resulting IMFs are narrowband in character
but not strictly monocomponent, potentially encapsulating in that narrow band both
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closely spaced harmonics, leading to the inability of the Hilbert spectrum to capture the
dual harmonic character of this signal. Unfortunately, the frequency resolution of EMD
cannot be readily refined.
7.5.6 Example 6: Amplitude Modulated Signal with Constant Frequency
In another example provided by Huang et al. (1998), the physical significance of the HSA
result is again called into question. In this case, an amplitude-modulated (AM) wave is
generated by
⎛ 2 ⎞
x = exp(−0.01t ) cos⎜ πt ⎟
⎝ 32 ⎠
(7.13)
which represents the impulse response function of a single-degree-of-freedom mechanical
oscillator with frequency of 0.0313 Hz and damping ratio of approximately 5%, shown in
Figure 7.8a. Although the signal is completely amplitude-modulated in theory, there is a
minor frequency modulation revealed upon applying the Hilbert transform, as shown later
in Figure 7.9b. Huang et al. (1998) argue that this result should be expected as amplitude
variations influence the bandwidth of a process, viewed in terms of traditional power
spectra as a spread of frequencies about the mean frequency. Based on that argument, the
authors contend that this spread of frequency may be manifested in time as a slight
deviation of the IF from the mean frequency of the process, in this case yielding
oscillations about a frequency of 0.0313 Hz (see Figure 7.9b). This spread of frequencies
about the IF at each instant in time is one means to account for the many neighboring
frequencies produced by AM, however, the IF should theoretically surface as the average
of the frequencies at each point in time, as discussed in Priestley (1988) and would be
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f [Hz]
f [Hz]
β [Hz]
(b)
(f)
(g)
t [s]
(c)
(h)
(d)
|W(a,t)|2
f [Hz]
f [Hz]
f [Hz]
x(t)
(a)
(e)
t [s]
f [Hz]
t [s]
FIGURE 7.8. Example 6: (a) amplitude-modulated cosine; (b) wavelet scalogram;
(c) WIFS; (d) wavelet scalogram with application of padding; (e) WIFS with
application of padding (zoom); (f) WIFS determined from wavelet phase with
application of padding (zoom); (g) wavelet estimate of HPBW with application of
padding; (h) 3D view of wavelet scalogram with application of padding
expected not to oscillate for this linear system. As the signal has no true frequency
modulation, how should the oscillatory component detected by the HT be interpreted?
On the other hand, the analysis by the continuous wavelet transform (fo=1 Hz),
shown in Figure 7.8b, captures the transient nature of the signal, with the reds denoting
the maximum energy content concentrated near 0.03 Hz. Examining the WIFS in Figure
7.8c illustrates that the extracted estimate of the IF (0.0315 Hz), though lacking accuracy
at the initiation of the signal due to an end effect, is within 1% of the theoretical
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f [Hz]
(a)
f [Hz]
(b)
t [s]
FIGURE 7.9. (a) Huang et al.’s (1998) Morlet
wavelet result; (b) Huang et al.’s (1998)
Hilbert spectrum (zoom)
frequency, shown by the dotted line. Beyond 400 seconds, the identification of the IF is
no longer accurate, as the signal energy is fading. Note, as shown in Figure 7.9a, that
Huang et al. (1998) similarly observed end effects anomalies in their wavelet analysis. As
discussed previously, end effects can pose a significant limitation to the quality of
wavelet analyses, particularly for short duration signals. To insure local preservation of
spectral characteristics, the signal in Equation 7.13 is padded using the reflective padding
scheme previously introduced. The resulting scalogram and WIFS are shown in Figure
7.8d-e. The use of this technique eliminates the poor performance at the initiation of the
signal and greatly enhances its accuracy at the conclusion of the signal, though there are
332
still some deviations due again to the fact that the signal, at that point, has nearly damped
out completely.
To further validate the ability of the wavelet to analyze this signal and
demonstrate the dual potential of the wavelet, the instantaneous frequency is also
identified using the derivative of the wavelet phase. As this often can be a more precise
means of identifying the instantaneous frequency, it is a useful exercise to see if the
wavelet can detect any physical influence of amplitude decay in the instantaneous
frequency. As shown in Figure 7.8f, the wavelet instantaneous frequency extracted from
phase takes on a constant value of 0.0313 Hz and when zoomed in to a scale of +/-1% of
the oscillator frequency, there is no evidence of oscillation. There is a slight deviation
early in the signal, corresponding to a residual byproduct of end effects. At the end of the
signal, the estimation quality rapidly degrades due to the difficulty of phase identification
once the signal energy is nearly completely damped out.
It was also demonstrated in previous examples that instantaneous bandwidth can
manifest evidence of nonlinearity or physically meaningful intrawave frequency
modulation. SDOF oscillators like the one considered here, the half-power bandwidth has
a unique relationship to the oscillator frequency and its damping. The HPBW of each
instantaneous spectra produced from the wavelet analysis is provided in Figure 7.8g and
demonstrates that this value holds relatively constant throughout the decay in the signal,
as expected, since the expression in Equation 7.4 represents an oscillator with constant
frequency and damping for which both the resonant frequency and bandwidth should not
vary with time. Note that at the beginning of the signal, the bandwidth suffers from a
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more visible inaccuracy, attributed to the fact that the bandwidth measure is far more
sensitive than the instantaneous frequency to end effects, as discussed in Chapter 4 and
demonstrated by several additional examples in Chapter 9. As discussed later in Chapter
9, even with the addition of padding, bandwidth measures within 3∆t of the beginning
and end of the signal can have some residual inaccuracy. For this analysis, that region
would encompass 68 seconds at the ends of the signals, clearly marked by the rounded
characteristic in Figure 7.8g. Neglecting these two regions, the bandwidth holds constant,
as expected. The sudden drop off beyond 450 s is again the consequence of attempting to
identify a signal for which has almost entirely damped out. Figure 7.9b similarly affirms
that the Hilbert Spectrum can also no longer identify frequency content beyond 200
seconds, again to the decay of signal energy.
Contrary to the HSA result, oscillatory frequency modulations are not reflected in
the wavelet HPBW measure, or in the instantaneous frequencies identified using either
the amplitude or phase of the wavelet. In fact, the presence of amplitude modulations in
this signal are ultimately observed in the three-dimensional perspective offered by the
wavelet scalogram in Figure 7.8h, where the amplitudes of wavelet coefficients reflect
the decay of energy in the signal. This information has been used in system identification
applications where nonlinearities in damping and stiffness are detected via changes in the
wavelet amplitude and phase, respectively, with time (Feldman, 1994a,b; Staszewski
1998). This example illustrates that the harmonic wavelet analysis will identify the true
frequency of the signal and does not identify an amplitude-induced FM in either its IF or
bandwidth, though identification of the amplitude modulation can be retained from the
amplitude of the wavelet coefficients.
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While the notion of intrawave frequency modulation may be physically
meaningful in some cases, in this example the presence of this phenomenon is not,
effectively indicating a slight nonlinearity in a linear the system, though it is noted in
Huang et al. (1998) that the magnitude of this intrawave modulation is only +/- 1.5%.
Still it must be clarified if this frequency modulation is due to physical changes in
frequency or due to a numerical byproduct of the HT. The analytic signal generated using
the HT will always provide a unique complex representation, however, as Boashash
(1992a) has stated, “whether or not it corresponds to any physical reality is another
question.” This indeed depends on the extent to which asymptotic signal assumptions are
met. In this case, the presence of frequency modulation in a constant frequency oscillator
can be deceiving and may actually be the consequence of a violation of asymptotic signal
assumptions. The definition of the instantaneous frequency in Equation 3.27 is only valid
if the envelope and phase are well separated. For an arbitrary signal in the form of
y ( t ) = a ( t ) cos( φ ( t ))
(7.14)
as in Equation 7.13, this implies that the Fourier transform of a(t) must be well separated
from and less than the Fourier transform of cos(φ(t)), as discussed in Chapter 3. This is
because the HT inherently selects the highest frequency component of the signal as the
complex phase term. If the Fourier spectrum of the oscillatory component of the signal is
not located at a frequency higher than and well separated from the envelope's spectrum,
then the Hilbert transform operation will be a result of overlapping and phase-distorted
functions. This will give rise to a waveform that can no longer be described by a purely
AM law, even though it was generated by an AM process (Rihaczek, 1966). In such
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x(t)
PSD [dB]
PSD [dB]
(d)
(a)
t [s]
t [s]
(b)
f [Hz]
(c)
f [Hz]
(e)
f [Hz]
(f)
f [Hz]
FIGURE 7.10. (a) Amplitude-modulated cosine in Example 6; (b) Fourier
spectrum of envelope a(t); (c) Fourier spectrum of phase cos(φ(t)); (d) Amplitudemodulated cosine in with envelope and phase well separated in frequency; (e)
Fourier spectrum of envelope a(t); (f) Fourier spectrum of phase cos(φ(t))
cases, "the HT and the analytic signal are not always interpretable in a way which is
physically meaningful and representative of physical phenomena" (Boashash, 1992a).
Examining the signal generated in Example 6 (Fig. 7.10a), a plot of the Fourier
spectrum of its exponential envelope and phase term at 1/32 Hz is shown in Figure
7.10b,c. In this case, it is clear that there is some overlap in these two spectra, indicating
that the envelope and phase may not be well separated in frequency and may cause
phase-distortions for the Hilbert transform. Figure 7.10d-f provides an example of a
signal with no significant spectral overlap between the amplitude and phase, as the cosine
frequency in Equation 7.14 is increased to 4 Hz. It is interesting then to examine the
implications of these overlapping envelope and phase spectra on the estimation of the
instantaneous frequency using the Hilbert transform. Figure 7.11b,c shows the estimated
instantaneous frequencies of these cases. The axis on the frequency coordinate is scaled
336
(c)
(b)
f [Hz]
(a)
t [s]
t [s]
t [s]
FIGURE 7.11. Instantaneous frequencies identified from the Hilbert transform: (a)
case where envelope and phase spectra overlap significantly; (b) case where
envelope and phase spectra overlap somewhat (Ex. 6), shown in Figure 7.8; (c)
case where envelope and phase spectra do not overlap significantly
to +/-1% of the true frequency in each case to provide an equivalent basis for comparison.
For further illustration, a case where the level of overlap is greater than in Example 6 is
also provided in Figure 7.11a by decreasing the cosine frequency in Equation 7.13 to 1/64
Hz. In this case, where the overlap is most significant (Fig. 7.11a), the level of this
intrawave modulation is more marked. As the overlap is lessened in Example 6 (Fig.
7.11b), the intrawave modulation is still present but reduced. In case of the higher
frequency signal in Figure 7.11c, the presence of this oscillation is hardly notable. It can
be inferred that the intrawave modulation in this case actually results from the Hilbert
transform not being able to clearly identify the phase and misinterpreting contributions
from the envelope as a result of their overlap in the Fourier domain.
In the case of this exponentially decaying envelope, by increasing the frequency
of the oscillatory term in Equation 7.14, the degree of overlap is not only being
minimized, but also the bandwidth of the system is also being increased. Note that the
product of the bandwidth and duration, or BT product, was introduced in Chapter 3 as a
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measure of the richness of the signal. The exercise in Figure 7.10 equally illustrates the
role of the BT product in the reliability of an IF measure produced by the HT. For signals
with small BT, e.g. a low frequency, well damped oscillator, the IF is difficult to identify
at an instant since there is no dominant frequency (Boashash, 1992a). This does not
preclude the use of the HT for small BT signals – it only raises concerns for the
interpretation of results.
In practice, it is desirable to examine asymptotic signals, which are characterized
by a finite duration, bandwidth and energy, but with large BT product; however, the term
“large” is quite ambiguous. Although, Slepian et al. (1961) determined that 99% of a
signal’s energy is typically preserved within the limits of BxTx > 5, Boashash (1992a)
suggested limiting analysis to signals with BxTx >10 such that the approximation error
resulting from assuming band and time limited functions is minimal. This will include, in
practice, signals used in communications and seismic applications. In Boashash (1992a)
it is shown that for a given signal, the analysis of short duration records (small BT) will
manifest significant leakage in the time-frequency energy density. Increasing the duration
(larger BT), reduces this effect so that the deviations from the true IF diminish.
Thus, the demonstrations in Figure 7.11 further illustrate that the estimated IF
approaches the anticipated result, with levels of IF oscillation decreasing, as the
bandwidth of the signals, and thereby the BT product, increases. Either the possibility of
spectral overlap between the phase and envelope or the implications of a small BT
product serve as viable explanations for the apparent “intrawave frequency” modulation
338
cited by Huang et al. (1998) and demonstrate that this characteristic is not physical but a
numerical byproduct of the Hilbert transform.
7.5.7
Example 7: Duffing Equation
Huang et al. (1998, 1999) explore a variety of classical nonlinear problems with distinctly
different frequency nonlinearities using continuous wavelets and HSA. The simplest of
these systems is the Duffing oscillator under harmonic excitation, in accordance with the
second-order differential equation
(
)
d 2x
+ 1 + ε x 2 x = γ cos ω t
2
dt
(7.15)
where ε, γ are constants and ω is the harmonic forcing frequency. The term in parenthesis
represents nonlinearity in the stiffness of the oscillator, which will lead to a frequency
that changes with amplitude. As discussed in Stoker (1966), Duffing (1918) noted that
the motions of this system were periodic but not simple harmonic, resulting in a period
that is “a unique function of amplitude”. The variation of frequency with time observed
in the Duffing oscillator is a physical reality due to the nonlinearity of the stiffness in
Equation 7.15 and has been identified by other authors using both Hilbert (Feldman,
1994a,b) and wavelet transforms (Staszewski, 1998). Huang et al. (1998, 1999)
investigated the solution for a strongly nonlinear case (Fig. 7.12a) using HSA and
identified 3 IMF components contributing to the system response, as shown in Figure
7.12b, for ε = -1, γ = 0.1 and ω = 1/25 Hz. The dominant component of the oscillator
displays marked intrawave oscillations of both long and short period, expanded upon in
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x(t)
(a)
f [Hz]
(b)
f [Hz]
(c)
t [s]
FIGURE 7.12. Example 7: (a) Duffing oscillator
under harmonic excitation; (b) Huang et al.’s (1998)
Hilbert spectrum manifesting 3 IMF components; (c)
Huang et al.’s (1998) Morlet wavelet result
Huang et al. (1998). Meanwhile, the low frequency component, represented by the IMF
near 0.04 Hz, is attributed to the forcing function (at 1/25 Hz). The lowest frequency
component is attributable to a slow wobbling of the phase. Huang et al. (1998) noted that
the Morlet wavelet could not capture this nonlinear behavior in terms of the intrawave
frequency modulations, though identifying the dominant frequency content in the region
near the intrinsic frequency of 0.124 Hz as shown by Figure 7.12c. Note however that
even this basic wavelet analysis by Huang et al. (1998) manifests some oscillatory
characteristic in the high frequency range, indicating a time-varying bandwidth and
340
x(t)
(a)
f [Hz]
(b)
|W(a,t=ti)|2
(d)
f [Hz]
(c)
(e)
f [Hz]
f [Hz]
Β [Hz]
(f)
t [s]
FIGURE 7.13. Example 7: (a) Duffing oscillator in free vibration, zoomed in
from 50 to 100 s; (b) wavelet scalogram; (c) WIFS; (d) wavelet instantaneous
frequency spectra at each time step plotted one atop another; (e) instantaneous
frequency identified by phase of WT; (f) instantaneous half-power bandwidth
demonstrating again that the wavelet’s ability to detect nonlinear behavior lies in its
bandwidth measure.
In order to expand upon these findings, the response of the system in Equation
7.15 is generated using a 4th order Runga Kutta simulation to determine if the wavelet is
sensitive enough to detect even subtler nonlinearities, embodied by a Duffing oscillator
with ε = -0.22, γ = 0.1. For a more basic understanding of the wavelet interpretation, the
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system is first released from the initial conditions of x(0) = 0 and x’(0) = 1 and allowed to
freely vibrate, as shown in Figure 7.13a. From the time series itself, the frequency of
oscillation can be estimated as 0.14 Hz. Applying the Morlet wavelet with fo=0.25 Hz, in
order to achieve a refined temporal resolution, yields a scalogram capturing the dominant
harmonic component, shown in Figure 7.13b. The extracted WIFS shown in Figure 7.13c
reiterates this finding through a single ridge near 0.12 Hz. The lack of accuracy in this
identified instantaneous frequency ridge can be justified in light of the reduced frequency
resolution for fo=0.25 Hz, which was conceded to enhance the temporal resolution.
Plotting the instantaneous spectra at each time step one atop another (Fig. 7.13d),
essentially collapsing the scalogram in time, confirms the presence of a single response
component, however the thickness of the profile indicates that there is some variation in
the amplitude, peak position and bandwidth. Although the maxima of the wavelet
scalogram cannot be used to directly identify the instantaneous frequency accurately for
such a fine time resolution analysis, the phase of the wavelet transform can still be used
to provide a more precise identification of instantaneous frequency. The IF determined
from the wavelet skeleton’s phase along the ridge in Figure 7.13c is provided in Figure
7.13e and affirms the presence of a frequency component near 0.143 Hz. The oscillation
of the phase instantaneous frequency reflects at the subtle nonlinearity of this system. The
findings in Figure 7.13e imply that the peaks of the instantaneous spectra in Figure 7.13d
are gradually shifting back and forth about 0.1431 Hz. By examining the instantaneous
half-power bandwidth, Figure 7.13f provides additional evidence of the subtle
nonlinearity of this system with oscillations occurring at twice the frequency of the
oscillation of the Duffing system. The fluctuations of the bandwidth measure describe the
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variation of the spread of frequencies about the instantaneous frequency in Figure 7.13e,
the source of the thickened line on the backside of the instantaneous power spectra. As
the red frames in Figure 7.13 help to demonstrate, the bandwidth, sensitive to signal
amplitude, takes on its lowest values in the signal’s troughs and peaks at the zero
crossings of the signal, where the Duffing oscillator is essentially a linear system, while
intermediate troughs occur at the signal maxima. As the envelope of the signal in Figure
7.13a is constant due to the lack of dissipative term, one may expect a constant
bandwidth in the response. However, the subtle bandwidth fluctuations are a result of the
nonlinearity in frequency and affirm the bandwidth sensitivity to even very mild
nonlinear phenomenon. It should be reiterated that if fo were chosen too large, much of
this behavior would be obscured, as also demonstrated previously in the Stokes wave
analysis.
Looking at the forced response of the same system for ω = 1/50 Hz in Figure
7.14a, the envelope of the signal manifests a low frequency oscillation as a result of the
forcing function. Inspection of the wavelet scalogram in Figure 7.14b reveals that the
warmest hues indicate the frequencies at which the signal concentrates, which again are
focused at 0.12 Hz due to the loss of frequency resolution in the scalogram discussed
previously. The bright blue band at the foot of the scalogram indicates the 0.02 Hz
forcing function. While the energy is clearly concentrated in the vicinity of the
oscillator’s dominant frequency, the higher frequencies of the scalogram manifest a
rippling indicative of fluctuations in the frequency content with time. The instantaneous
frequencies were identified from the ridges of the transform and are shown in Figure
7.14c. Two dominant components were observed: one near 0.12 Hz and one
343
x(t)
(a)
f [Hz]
(b)
|W(a,t=ti)|2
(d)
f [Hz]
(c)
(e)
Β [Hz]
f [Hz]
f [Hz]
(f)
f [Hz]
(g)
t [s]
FIGURE 7.14. Example 7: (a) Duffing oscillator in forced vibration; (b)
wavelet scalogram; (c) WIFS; (d) wavelet instantaneous frequency spectra at
each time step plotted one atop another; (e) instantaneous frequency of high
frequency component identified by phase of WT; (f) instantaneous halfpower bandwidth of high frequency component; (g) estimate of
instantaneous frequency by Hilbert Transform
344
corresponding to the forcing frequency near 0.02 Hz. Contrary to Figure 7.13b, the ridge
associated with the nonlinear oscillator now manifests some fluctuation. The same
overlay of instantaneous spectra in Figure 7.14d reveals that a more marked spread in
bandwidth, peak frequency and amplitude of the spectral peak than in its counterpart in
Figure 7.13b, indicative of more prominent time-varying characteristics. Turning again to
the instantaneous frequency estimate from the wavelet phase in Figure 7.14e, the range of
oscillation of the instantaneous frequency is identified more precisely in the vicinity of
0.14 Hz, as before. Note that the wavelet instantaneous frequency estimate, grounded in
Fourier harmonics, manifests a smooth regular periodicity. Again recalling that the
wavelet analysis fits small waves to the data, the instantaneous frequency estimate is
more representative of the best-fit frequency over some short time interval but may not
fully capture subcyclic characteristics. Turning to the instantaneous bandwidth in Figure
7.14f, a different perspective with a far more oscillatory characteristic is evident. The red
box denotes a region of downshift, during which the local signal mean shifts to lower
amplitudes. Each downshift is followed by an upshift. These trends occur in a periodic
manner as a consequence of the forcing frequency, thus the duration of downshift region
is half the forcing period or 25 s. The initiation and termination of the downshift are
marked by troughs in the wavelet instantaneous frequency in Figure 7.14e and the
deepest troughs of the instantaneous bandwidth in Figure 7.14f. This pattern repeated in
the bandwidth is a consequence of the amplitude modulation arising from the forcing
function, while the higher frequency fluctuations spanning the downshift region reflect
the nonlinearity of frequency in the signal. Note that both the frequency and bandwidth
qualitatively manifest local symmetry over the downshift region. This result can be
345
compared to the instantaneous frequency estimated via the Hilbert transform, shown in
Figure 7.14g. This result shows more irregularity when compared to the wavelet
instantaneous frequency in Figure 7.14e, as expected. The wavelet instantaneous
frequency is characteristically smooth as discussed above, being a best-fit local mean to
the frequency at that time. For the wavelet, any fluctuations or spread about this mean are
carried in the bandwidth in Figure 7.14f. The Hilbert result does not separate these two
contributors of frequency evolution and provides an instantaneous frequency estimate
whose peaks and troughs shift and lean, though in a repeating pattern, as a result of
intrawave fluctuations. The appropriateness of either representation depends entirely on
the perspective desired.
This example again reiterates the distinction between these two techniques: the
HSA will in general detect these subtle changes in frequency due to some nonlinearity,
essentially identifying subcyclic frequency changes, while the Morlet wavelet’s
instantaneous frequency will only detect changes in the instantaneous frequency in its
truest mean sense, as it locally fits windowed sinusoids to the data and is thus capable of
detecting only nonlinearities evolving over entire cycles of oscillation or following
significant changes in amplitude, which will be referred to herein as supercyclic
oscillations. One period of oscillation in the Duffing oscillator corresponds to a marked
amplitude change, easily detectable by the Morlet wavelet’s instantaneous frequency
extracted from the phase.
However, the rippling manifested in the scalogram, particularly in the high
frequency range, suggests that the wavelet is detecting some of the subcyclic oscillations
346
of the Duffing oscillator, not through its ridges or instantaneous frequency, but through
its spectral bandwidth, as introduced in Example 5. The bandwidth fluctuations, being
more sensitive than instantaneous frequency, are not as precise near end effects regions,
as discussed in more detail later in Chapter 8. Even so, they capture repeating patterns of
frequency spread with time. Further, they demonstrate a secondary fluctuation in the
bandwidth with a periodicity that corresponds to the time between a signal peak and its
subsequent trough – a subcyclic phenomenon. Thus, it appears that while the wavelet’s
instantaneous frequency captures only supercyclic oscillations and mean behaviors of
frequency over small windows, the additional insights provided by the bandwidth affirm
the deviations from the instantaneous frequency, affirming the presence of subcyclic
phenomenon or a frequency that continuously varies with amplitude for the same reasons
as discussed following the Stokes wave examples.
7.5.8 Example 8: Lorenz Equation
The Lorenz equation, initially proposed to study deterministic non-periodic flow, has
become a fundamental system for the investigation of chaos and was considered in
Huang et al. (1998) as another classic nonlinear system for investigation using HSA. The
system is described by
x& = −σx + σy
y& = rx − y − xz
z& = −bz + xy
(7.16)
where s, r and b are positive constants, assumed to be 10, 20 and 3, respectively, for the
purpose of this example. The system is released from its initial position of (10, 0, 0),
resulting in the x-component response shown in Figure 7.15a, illustrating the transient
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x(t)
(a)
t [s]
(b)
f [Hz]
f [Hz]
(c)
t [s]
t [s]
FIGURE 7.15. Example 8: (a) Lorenz equation x-component; (b)
Huang et al.’s (1998) Hilbert spectrum of Lorenz equation, xcompnent; (c) Huang et al.’s (1998) Morlet wavelet result
characteristics of the system. The Hilbert spectrum generated by Huang et al. (1998) is
provided in Figure 7.15b and reveals the characteristic intrawave frequency modulation
observed in other non-linear systems as a result of subcyclic frequency variations. The
Hilbert spectrum also manifests a low frequency component, which rapidly drops from 1
Hz and lingers around 0.1-0.2 Hz. This mode corresponds to the rapid transient drop
occurring in the first few seconds of the Lorenz response. After this transient period, the
oscillator tends to behave as a damped nonlinear system. Note the decaying of the
instantaneous frequency indicating supercyclic FM due to the damping characteristic as
well as the oscillations about 1.4 Hz due to subcyclic nonlinear behavior. Linearization of
the system in Huang et al. (1998) yielded a dominant frequency of about 1.46 Hz,
consistent with the main frequency about which intrawave modulations occur.
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The continuous wavelet analysis by Huang et al. (1998) shown in Figure 7.15c
captures the transient nature of the signal, but the frequency resolution may not be
sufficient to disclose the nonlinearities present. The revisited Morlet wavelet analysis of
the Lorenz x-component response (Fig. 7.16a) is shown in Figure 7.16b. While the
scalogram indicates that the wavelet has captured the transient behavior with the intense
red patches, it is the wavelet instantaneous frequency spectrum (Fig. 7.16c) that truly
provides clear representation of the behavior. Consistent with the HSA in Figure 7.15b,
the Morlet wavelet with fo=0.5 Hz, chosen for more precise temporal resolution, was
capable of detecting the marked frequency shifts due to the transient behavior in the first
few seconds of the signal. This is characterized by a shift in the low frequency
component from about 0.75 Hz to 0.019 Hz. After this transition range, the wavelet
detects a bi-modal response, with the low frequency response near 0.019 Hz and the
oscillator’s response at about 1.4 Hz, near the 1.46 Hz theoretical prediction of the
linearized system. The estimation of instantaneous frequency of this mode becomes
unreliable beyond 10-15s, as the signal’s energy decays, shown by the diminishing light
blue patch in the scalogram. This observation is consistent with the HS in Figure 7.15b,
which indicates that the instantaneous frequency estimate about 1.4 Hz is scarcely
detected beyond 10 seconds, though the lower frequency response is still clearly present.
The subcyclic oscillations about 1.4 Hz, occurring essentially in the decaying component
of the response, are subtly evident in the WIFS in Figure 7.16c, apparently becoming
more pronounced as the signal dissipates. To more closely inspect these subcyclic
oscillations, the instantaneous frequency of this component is also identified from the
wavelet phase. The result, shown in Figure 7.16d demonstrates that subtle oscillations are
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x(t)
(a)
f [Hz]
(b)
f [Hz]
(d)
Β [Hz]
f [Hz]
(c)
(e)
t [s]
FIGURE 7.16. Example 8: (a) Lorenz equation x-component;
(b) wavelet scalogram; (c) WIFS; (d) instantaneous frequency
of high frequency component identified by phase of WT; (e)
instantaneous half-power bandwidth of high frequency
component
detected by the wavelet phase, though not of the magnitude detected in HSA. These
oscillations grow in amplitude slightly, however, this is not a physical reality but rather a
bi-product of the increasing difficulty in estimating the instantaneous frequency
accurately from the phase as the signal’s energy rapidly diminishes. The amplitude of
wavelet coefficients during this decaying phase of the higher frequency component
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initiate at a maximum value of around 8.5 and fall beneath 10% of this value by the tenth
second, becoming essentially zero by the twentieth second.
The dual characteristic of the continuous wavelet transform allows the
investigation of instantaneous spectra, taken as slices of the wavelet scalogram along the
time axis. These spectra provide snapshots of the system’s frequency domain behavior.
Critical points in the response of the Lorenz equation’s x-component are labeled in
Figure 7.17 by asterisks, and their associated instantaneous spectra are provided. The first
three points are part of the rapid transient decay. Their instantaneous spectra reveal the
predominantly monocomponent nature of this transient region, with the broad spectral
peak shifting from near 1 Hz to a lower frequency value near 0.5 Hz. By the fourth point,
the system has entered into an essential free vibration about 1.4 Hz accompanied by the
persistent low frequency component. The presence of both modes continues through
points 5 and 6, though the magnitude of the second peak associated with the free
vibration oscillations is steady decreasing as the system’s energy is dissipated.
Simultaneously, the low frequency component approaches DC and becomes highly
concentrated, reflecting the permanent offset of the system as the response decays. An
investigation of the half-power bandwidth of the second peak is provided in Figure 7.16e,
affirming again the presence of nonlinearity in the frequency spread at each instant in
time, with a periodicity similar to that observed in the phase-identified instantaneous
frequency. It should be noted that the diminished values of the bandwidth at the
beginning of the signal is a consequence of enhanced sensitivity in the end effects region,
discussed more fully in Chapter 8. Again, the WIFS and wavelet bandwidth concur with
the findings of HSA: observing an oscillatory characteristic in the high frequency
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25 6
3
1
|W(a,t=ti)|2
x(t)
1
7
4
f [Hz]
t [s]
f [Hz]
f [Hz]
6
7
|W(a,t=ti)|2
|W(a,t=ti)|2
5
4
|W(a,t=ti)|2
f [Hz]
f [Hz]
|W(a,t=ti)|2
3
|W(a,t=ti)|2
|W(a,t=ti)|2
2
f [Hz]
f [Hz]
FIGURE 7.17. Example 8: Lorenz equation, x-component,
instantaneous spectrum from wavelet transform at critical points in
the response
component of this nonlinear system and identifying the transient behavior of the constant
offset.
7.5.9 Example 9: Rössler Equation
Huang et al. (1998) also investigated the Rössler Equation for the famous period doubling
event described by
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x& = − ( y + z )
y& = x +
1
y
5
z& =
1
+ z( x + µ )
5
(7.17)
where µ = 3.5. The system simulated by a first-order forward difference technique from
the initial conditions {x(0) = -4, y(0) = 4, z(0) = 0} is shown in Figure 7.18a. The Hilbert
spectral analysis of the x-component of the Rössler system, as performed by Huang et al.
(1998), is provided in Figure 7.18b. Note that the EMD of the data yielded 2 meaningful
IMF components, one oscillating between 0.17 and 0.25 Hz, representing the dominant
nonlinear characteristic of this system. The lower frequency component, near 0.1 Hz, is
assumed to be the result of some numerical error, as it is quite low in energy and is
surfacing in a system known to have only two dominant scales.
Huang et al. (1998) states that “to represent such a [system] with either Fourier
spectrum or wavelet analysis, one would need many harmonics.” Indeed, a continuous
wavelet analysis may represent globally this system by a series of harmonics if the
temporal resolutions are insufficient to resolve the nonlinearities, as shown by Figure
7.18c. Inspecting the signal shown again in Figure 7.19a, there are large amplitude peaks
occurring about every 11 seconds, indicating that there is some frequency content in the
vicinity of 0.1 Hz. Temporally, smaller peaks occur approximately 5 seconds after a large
peak, so the wavelet chosen must not have a time resolution greater than 5 seconds in
order to detect the changes in the signal with time. Wavelet analyses can often be
performed in stages. In the first stage, it is often desirable to identify components within
the signal by choosing a wavelet with sufficiently fine frequency resolution. As the
dominant peaks have a frequency in the vicinity of 0.1 Hz, one may seek a wavelet with a
resolution 10% of this value, or 0.01 Hz. This can be achieved using fo=5 Hz, though the
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x(t)
(a)
f [Hz]
f [Hz]
(c)
(b)
t [s]
t [s]
FIGURE 7.18. Example 9: (a) Rössler equation x-component; (b) Huang et al.’s
(1998) Hilbert spectrum; (c) Huang et al.’s (1998) Morlet wavelet result
temporal resolution for this wavelet is about 7 seconds and thus will not capture the
variations which occur every 3-4 seconds. Using this fine frequency resolution first, in
stage 1 of the analysis, two distinct components are identified at around 0.09 and 0.17
Hz, demonstrated in Figures 7.19b-c. This information then defines two frequency ranges
over which to apply a more temporally refined analysis in stage 2. The first component
near 0.1 Hz was cited by Huang et al. (1998) as some numerical artifact from the
simulation of the system. As this component does not manifest marked temporal variation
in the refined analysis, it will not be included in subsequent discussions. However, it will
be shown that the component near 0.17 Hz does reveal interesting phenomena in a
refined analysis. Again, to track the time-evolving amplitude changes, a wavelet with fine
time resolution must be chosen. For fo=0.25 Hz, the temporal resolution is enhanced to
approximately 1.75 s. Using this wavelet to analyze the second component of the system
as stage 2, the scalogram in Figure 7.19d is obtained. Note that the color contours of the
wavelet scalogram mimic the same oscillatory peaking of the frequency observed in the
Hilbert spectrum in Figure 7.18b. The WIFS is generated in Figure 7.19e and reveals that
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x(t)
(a)
f [Hz]
|W(a,t=ti)|2
(c)
f [Hz]
(b)
f [Hz]
(d)
Β [Hz]
(e)
(f)
t[s]
FIGURE 7.19. Example 9: (a) Rössler equation x-component; (b) WIFS for
analysis stage 1; (c) sample instantaneous spectra; (d) wavelet scalogram for
analysis stage 2; (e) WIFS for analysis stage 2; (f) instantaneous half-power
bandwidth for analysis stage 2
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the energy is primarily concentrated in an oscillatory pattern between 0.1375 and 0.1665
Hz, with a period of oscillation of 11.4 seconds, capturing the period of the large peaks in
the time series.
As the scalogram fluctuations in the high frequency range indicate some variation
of energy in time, a half-power bandwidth analysis is performed as part of stage 2 on this
second component. The results, shown in Figure 7.19f, clearly demonstrate the behavior
identified by Huang et al. (1998) in Figure 7.18b, with large peaks interlaced with
smaller, rounded humps. The period between the major peaks in bandwidth is 11.4
seconds, corresponding to the period of the large peaks in the signal. Approximately 5.5
seconds after each major peak in the HPBW, a smaller peak follows, as also observed in
the time series. The period between the smaller peaks of the HPBW is also 11.4 seconds.
Thus the dual identification of the instantaneous frequencies and bandwidth from the
wavelet transform illustrates the ability of the Morlet wavelet to characterize this
nonlinear behavior, albeit differently than the HSA, and dispute the claim by Huang et al.
(1998) that “no other methods can match the resolution power displayed here.” As
demonstrated in previous examples, while the instantaneous frequency can signal the
presence of nonlinear behavior in response to marked amplitude changes in the signal, it
does so in a more averaged sense. It is the sensitivity of the instantaneous bandwidth that
detects the presence of waves of neighboring scales indicative of a deviation from a
perfect sinusoid – a trademark of the nonlinear system.
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7.5.10 Example 10: Random Wave Data
While the wavelet analyses presented thus far have considered a number of nonlinear and
nonstationary systems, Huang et al. (1998) extended the time-frequency analyses to
measured wave data. While the intent of this chapter is not to provide such an exhaustive
array of examples, the analysis of measured wave data was included in the discussions of
the Stokes wave. To enrich this discussion further, the analysis of experimentally
observed random sea waves is presented herein. The waves were generated by a
JONSAWP random excitation with amplitude of 64 cm. The power spectrum of the
excitation affirms discernable frequency content up to approximately 2 Hz, although this
broad peak focuses at approximately 0.4 Hz. The resulting waves at one measuring
station are provided in Figure 7.20a for 200 seconds of the total measured record. From
inspection of the time series, the fundamental period of oscillation is approximately 2
seconds, though the wave profiles are not smooth, indicating the presence of other
frequencies or nonlinearities in the data. A continuous wavelet analysis with fo=1 Hz
produced the scalogram shown in Figure 7.20b. The scalogram reflects several pockets of
intense energy bursts, associated with high amplitude events in the data, and concentrated
at around 0.5 Hz. The presence of lighter hues fading into the high frequency range again
suggests the detection of a distribution of energy beyond the fundamental observed
frequency. Ridge extraction from the wavelet modulus revealed up to three local maxima
for any given instantaneous spectrum. The maxima take on the highest values in the
vicinity of 0.5 Hz, as shown in red on the WIFS in Figure 7.20c, accompanied by
intermittent lower amplitude components in a relatively lower and higher frequencies,
shown in light green and yellow. Individual analyses of the occurrence of the three ridges
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x(t) [ft]
(a)
f [Hz]
(b)
f [Hz]
(c)
t [s]
FIGURE 7.20. Example 10: (a) measured random wave
data; (b) wavelet scalogram; (c) WIFS for up to three
ridges (red → green indicates highest to lowest energy
components at each time step)
are provided in Figure 7.21. Figure 7.21a displays the instantaneous frequency associated
with the simplest harmonic representation observed in the signal – for which there is only
one dominant oscillatory component near 0.5 Hz, typified by the instantaneous spectrum
to the right. The frequency content of the signal often shifts then to a bi-modal
characteristic, centered around 0.5 Hz, as shown in Figure 7.21b, but alternating its
dominant peak between approximately 0.4 and 0.6 Hz, as shown by the two example
instantaneous spectra. The occurrence of a third peak is usually an intermittent
phenomenon of relatively lower energy and accompanies the dominant presence of the
same two harmonics centered near 0.5 Hz. It is important to reiterate that wavelet
instantaneous spectra, when viewed in tandem with the wavelet instantaneous frequency
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f [Hz]
|W(a,t=ti)|2
(a)
f [Hz]
f [Hz]
|W(a,t=ti)|2
(b)
f [Hz]
|W(a,t=ti)|2
f [Hz]
(c)
f [Hz]
f [Hz]
t [s]
FIGURE 7.21. Example 10: instantaneous frequency identified from (a) single
mode spectra; (b) bi-modal spectra; (c) multi-mode spectra (red → green indicates
highest to lowest energy components at that time step), with example of each
instantaneous spectra class shown at the right
spectrum, serve as a microscope for studying the evolution of multiple harmonic
components within the response. In particular, the alternating characteristic of the two
dominant components centered near 0.5 Hz represents a temporal variation of the
fundamental wave frequency that would be obscured in traditional Fourier analysis. This
observation, coupled with the intermittent characteristics, further highlights the richness
of the energy distribution in the wave profile, similar to the findings in Huang et al.
(1998).
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7.6 Discussion: Processing Concerns and Performance in Noise
While the examples provided herein have reassessed the performance of the continuous
wavelet transform and the Hilbert transform with Empirical Mode Decomposition for a
number of nonlinear and nonstationary systems, these results are not achieved without a
proper understanding of each approach. For example, the successful application of the
Hilbert transform is predicated on the IMFs satisfying certain narrowband and symmetry
conditions. The quality of the IMFs produced by EMD is dependent upon the spline fit
procedure employed, particularly near the ends, where cubic spline fits have wide swings
that may eventually propagate inward and corrupt the entire data span. Corrective
measures such as the inclusion of characteristic waves are discussed in Huang et al.
(1998) and should be applied as needed to insure optimal performance. The use of spline
fitting or oversampling of the data will additionally be required to determine the
derivatives of the phase with sufficient accuracy. This is true of the wavelet analysis as
well if the phase is used to estimate the instantaneous frequencies, though in the
examples provided it is often sufficient to use the maxima of the transform for this
identification, alleviating the need for any differentiation.
The sifting process used to generate the IMFs should also be used with care, since
when applied in excess, it can completely obliterate meaningful amplitude modulations
and yield purely frequency-modulated components. Thus, the user of the EMD approach
should be well versed in determining when to terminate the sifting process and the
suitability of the resulting IMFs, as well as in distinguishing if any corruptions by the
spline fits are present.
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By the same token, the users of continuous wavelet transforms must be cognizant
that the result of their analysis relies heavily on the parent wavelet employed. As stated
by Lewalle (1998) “Each wavelet shape is selected by the user – as is the conventional
sine wave [of Fourier Analysis] -, and this limits the type of questions that can be
answered.” Thus the choice of an appropriate parent wavelet will vary in each application
and will greatly influence the results, though the Morlet wavelet is a natural choice for
any analysis seeking to replicate local Fourier perspectives. Further, “the freedom to
choose between wavelets implies the responsibility to interpret the results accordingly”
(Lewalle, 1998). The poor interpretation of Morlet wavelet results in Huang et al. (1998)
was largely responsible for many of the misleading statements surrounding wavelet
performance. Specifically, when using the Morlet wavelet, to completely exploit the
resolution capabilities, users should make careful selections of the central frequency fo
within the framework presented in Chapter 4. A lack of understanding of the durations of
the dilated time and frequency windows of the analysis can yield misleading results and
may have in part led to the poor results noted in the literature. Though Huang et al.
(1998) did not report the exact central frequency used in their Morlet wavelet analysis,
statements referring to the Morlet wavelet as “Gaussian enveloped sine and cosine wave
groups with 5.5 waves” would hint at the use of a central frequency near 5/(2π), the
default value in the MATLABTM Wavelet Toolbox. In this form, the undilated Morlet
wavelet would have only a few cycles of oscillation in the Gaussian window, as shown in
Figure 7.22a, clearly not appropriate for the wide array of signals analyzed here. In fact,
the discussion of the Stokes wave highlighted the fact that a high frequency resolution
wavelet analysis begins to approach the Fourier harmonic analysis and loses much of its
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(a)
Re[g(t)]
Re[g(t)]
(b)
t [s]
t [s]
FIGURE 7.22. (a) Real component of Morlet wavelet basis function for
fo=5/(2π) Hz; (b) real component of Morlet wavelet basis function for fo=5 Hz
temporal sensitivity, while a lower frequency analysis can uncover the nonlinear
characteristic. On the other hand, as demonstrated in Example 5, for the two closely
spaced waves, a higher value of central frequency was necessary to separate the two
modes. These higher values of central frequency permit improved frequency resolution
by packing more waves into the Gaussian window, as shown in Figure 7.22b. Therefore,
it should be noted that the value of this central frequency was changed throughout this
chapter and this work in order to properly tune the parent wavelet. As such discussions
are not mentioned in Huang et al. (1998), it is assumed that this value was not changed
for each example in that study, a fact reflected by the poor wavelet results in that work.
Further, to obtain the wavelet instantaneous frequency spectrum, users must
extract the ridges of the transform. By visual inspection, one can clearly discern the
highest energy in the contour map, but to obtain a skeleton plot comparable to the Hilbert
spectrum, ridges must be extracted, most easily by using the local maxima of the modulus
of the transform. In cases where excessive noise induces spurious local maxima, the use
of more sophisticated extraction techniques is required (Carmona et al., 1998). These
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approaches use the smoothness of the ridge as an additional condition in the minimization
scheme and require additional experience and programming, as discussed in Chapter 4.
Finally, the examples provided in this chapter have not explored either
technique’s performance in the presence of noise. Huang et al. (1998, 1999) did examine
several sets of measured data from earthquake, wind and wave events, which can be
assumed to contain some noise, from which the HSA was able to extract IMF
components and their instantaneous frequencies. However, in general, the performance of
HSA in noise is dictated by approach used to accomplish the differentiation required to
obtain the instantaneous frequency from the phase and the ability of the EMD process to
separate noise from the signal. The differentiation scheme is particularly crucial in the
accurate identification of the instantaneous frequency. A comparison of the performance
of several of these differentiation approaches in the presence of noise was conduced by
Boashash (1992b) and revealed that some approaches using the phase information (e.g.
central difference) may have considerable difficulty in noisy situations, requiring the use
of more sophisticated approaches such as Maximum Likelihood Estimators. However, all
of these discussions are relevant only to the Hilbert transform itself and not the overall
Hilbert Spectral Analysis. It is possible that the inclusion of EMD prior to the application
of the HT may have enhanced the performance discussed by Boashash (1992b). As a
result, these findings do not necessarily demonstrate a shortcoming of HSA.
On the other hand, approaches that utilized the peaks in the modulus of timefrequency distributions such as the STFT or WVD to determine the instantaneous
frequency were some of the most accurate in noisy situations (Boashash, 1992b),
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supporting the use of this approach for the continuous wavelet transforms in this study.
This point was also made by Carmona et al. (1997), who indicated that the phase might
be difficult to control in noisy situations, motivating the use of the wavelet modulus for
the extraction of ridges. Delprat et al. (1992) illustrated the ability of wavelet ridges to
identify a bat sonar signal embedded in noise with amplitude comparable to the signal
itself. This theory of wavelet ridges was also used by Staszewski (1997, 1998) to examine
the ability of wavelets to perform system identification by adding known levels of white
noise to simulated structural responses and found the technique to be quite robust, even in
the often problematic estimation of structural damping. The robustness of time-varying
frequency system identification by wavelets in the presence of noise was also quantified
by Ghanem & Romeo (2000), while the use of wavelets for cleaning noisy observations
was previously advocated by Coca & Billings (1997) and Gurley & Kareem (1999).
7.7 Discussion: A Fundamental Difference
The examples presented in this chapter have highlighted a fundamental difference
between the wavelet analysis and that provided by the HSA: the manner in which
physically meaningful intrawave or subcyclic frequency modulations are identified. The
concepts of subcyclic oscillations, denoting changes in frequency that occur within a
single cycle of oscillation, and supercyclic oscillations, denoting changes in frequency
that occur over the course of one or more cycles or due to rapid changes in amplitude, are
introduced to help explain the performance of the continuous wavelet transform and the
Hilbert spectral analysis. As exercises in nonlinear systems have indicated, the presence
of intrawave modulation may be rationalized as a physical phenomenon by both
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techniques. In the wavelet interpretation of these systems, the fitting of wavelets or small
waves of a given scale (frequency) tends to capture local mean trends in the instantaneous
frequency of the system. For systems undergoing significant amplitude fluctuations or
marked supercyclic oscillations, which would induce significant frequency variations,
such as in the Duffing, Lorenz and Rössler examples, the wavelet instantaneous
frequency captured fundamental supercyclic frequency modulations; however the subtle
characteristics of intrawave frequency variation are not held in this measure. As Priestley
(1988) indicated, the instantaneous frequency is an averaged measure of the frequencies
present at that instant in time. So for the Morlet wavelet, it is the frequency of the best-fit
sinusoid to the data over a shortened time window. However, sinusoids at neighboring
frequencies may also show some similitude with the signal over this same interval,
indicating oscillations that are not perfectly sinusoidal but deviate in some way. This
nonlinearity represented by subcyclic deviations must locally be treated by a summation
of neighboring harmonics. The spread of these neighboring frequencies is captured by the
instantaneous bandwidth measure presented in this chapter. When viewed in totality, the
wavelet instantaneous frequency represents an averaged sense of the frequency of the
system with time, and the bandwidth or deviation from this mean value, indicative of
evolving fluctuations in signal frequency content. As demonstrated in the case of Stokian
waves, these nonlinearities can be so subtle that they often only reveal themselves in the
wavelet’s instantaneous bandwidth. Thus the Hilbert instantaneous frequency captures
both super and subcyclic oscillations simultaneously, while the wavelet separates the two
classes of oscillations, with its instantaneous frequency detecting the presence of any
supercyclic oscillations and the instantaneous bandwidth capturing the subcyclic features.
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7.8 Conclusions
This chapter revisited many of the continuous wavelet examples used to establish the
efficacy of the Hilbert Spectral Analysis, not to advocate the use of one over the other,
but rather to represent both in a fair light and dispel some of the misconceptions
surrounding continuous wavelets for these applications. To provide a fair comparison, a
wavelet instantaneous frequency spectrum was introduced, which plots the wavelet
estimates of instantaneous frequency as a function of time. This perspective produces the
same skeleton plot as the Hilbert spectrum, which also must estimate instantaneous
frequency. Throughout the examples provided using this representation, the judicious
choice of the central frequency in the Morlet wavelet is reiterated, as improper temporal
resolution of the wavelet is shown to produce results that approach a traditional Fourier
analysis, while refined temporal resolutions were capable of identifying nonlinear and
nonstationary signal characteristics.
While the physical meaning of the HSA result was questioned in two examples
involving closely spaced cosine waves and an amplitude modulated, constant frequency
oscillator, the two approaches provided comparable evidence of nonlinear behavior for a
number of classical examples. However, this evidence was presented in distinctly
different manners: the instantaneous frequency of Hilbert spectral analysis detects both
subcyclic and supercyclic frequency modulations, while the wavelet instantaneous
frequency can only detect supercyclic frequency characteristics and instead relies on the
additional measure of the instantaneous bandwidth to provide subcyclic information. This
finding is expected, as the continuous wavelet fits small waves over a local window in
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time. The instantaneous frequency will in this case truly correspond to the scale
producing the best local fit of the wavelet to the data. However, this analysis is a local
summation of wavelets and therefore the number of additional wavelets at neighboring
scales required to truly fit the data also provides meaningful information on the spread of
frequencies about this instantaneous or best fit mean. The instantaneous bandwidth
measure provides this information and the key to uncovering subcyclic phenomena.
Therefore, the selection of one approach over the other is entirely dependent on the
perspective desired.
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CHAPTER 8
WAVELET SYSTEM IDENTIFICATION IV: FREQUENCY AND DAMPING
ESTIMATION FOR CIVIL ENGINEERING STRUCTURES
8.1 Introduction
As motivated previously, the dual time-frequency character of wavelet transforms allows
adaptation of both traditional time and frequency domain system identification
approaches to examine non-linear and non-stationary data. The adaptations of spectralbased approaches in this research were presented in Chapters 5, 6 and 7. In particular, the
coherence discussions in Chapter 6 foreshadowed some of the unique processing
challenges associated with this new analysis domain. Similarly, when utilizing timedomain information from wavelet coefficients in order to conduct more precise system
identification of dynamic properties, a number of the processing challenges associated
with wavelets, introduced and largely addressed in Chapter 4, have increasing relevance.
Although such challenges did not surface in prior applications concerned with
mechanical systems, characterized by higher frequency, broader-band signals, the
transition to the time-frequency domain for the analysis of Civil Engineering structures
highlighted the need to understand more fully the issues chronicled in Chapter 4,
especially for the popular Morlet wavelet. In particular, as these systems may possess
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longer period motions and thus require finer frequency resolutions, the specific impact of
end effects becomes increasingly apparent. This chapter discusses these considerations in
the context of the wavelet’s multi-resolution character and includes guidelines for
selection of wavelet central frequencies, highlights their role in the complete modal
separation measure introduced in Chapter 4, and quantifies their contributions to end
effects errors and their residual impacts on estimated instantaneous frequency and more
markedly, damping.
8.2 Previous Applications of Wavelets for System Identification
The Morlet wavelet and other classes of discrete and continuous wavelets have been
applied to a variety of problems ranging from image and acoustic processing to fractal
analysis. This study has showcased a number of these applications in the context of Civil
Engineering and the adaptation of wavelet transforms to situations where Fourier
transforms were traditionally used to define quantities of interest. Meanwhile, the more
rigorous application of wavelets to system identification of mechanical systems is still
advancing, but shows great promise thanks to the suite of parent wavelets available and
flexibility inherent in the wavelet transform itself, allowing the use of many traditional
time and frequency domain-based approaches.
In terms of analysis stemming from the frequency domain, the instantaneous
spectrum introduced in Section 4.4 can be utilized in more traditional frameworks for
system identification via wavelet-based frequency response functions (Hartin, 2001;
Staszewski & Giacomin, 1997; Kyprianou & Staszewski, 1999), as well as the coherence
applications in Chapter 6 and the general applications system identification in Chapters 5
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and 7. A number of other researchers have also noted this dual identification potential,
integrating the wavelet transform with number of system identification schemes
traditionally based upon Fourier transforms, e.g. Coca & Billings (1997), Ghanem &
Romeo (2000). Huang et al. (1994) used its localization and multi-resolution properties to
identify the mass, stiffness, and damping matrices from measured accelerations,
velocities, displacements and ground accelerations of MDOF systems to illustrate the
ability of the wavelet-based approach to accurately identify stiffness and damping from
short earthquake time histories, even in the presence of noise. Robertson et al. (1998a)
used the discrete form of the transform to extract impulse response functions from
measured system inputs and response. In comparison to traditional Fourier-based
approaches, the DWT-based approach provided superior results, even with limited
ensembles under various types of excitations. A second study then used this approach
along with the Eigensystem Realization Method (ERA) to identify the dynamic properties
and mode shapes of the system subjected to various excitations (Robertson et al., 1998b).
It should be noted that in the course of these studies, various parent wavelets were
employed in continuous and discrete forms of the transform.
As discussed previously, the analysis of free vibration or impulse response
function (IRF) from a structure serves as one of the simplest means to identify the
frequency and damping in the time domain. In the case of single degree of freedom
(SDOF) structures, this identification may be easily conducted using techniques such as
the logarithmic decrement, the Hilbert transform in the context of analytic signal theory,
or a least squares fit of the decay curve. However, in cases where multiple degrees of
freedom (MDOF) are participating, the identification becomes more involved and
370
necessitates the use of bandpass filtering or more advanced MDOF identification
schemes. Recognizing the ability of wavelet transforms to decouple multi-component
signals and their dual potential to allow analysis within the time domain, a number of
researchers began applying this technique in various forms for analysis of impulse or free
vibration response, e.g. Hans et al. (2000) and Lamarque et al. (2000). These concepts
were overviewed in Section 3.7.4 within the broader context of a discussion on wavelet
ridges and analytic signal theory in Section 3.7.
Furthermore, as the damping and natural frequency for a nonlinear mechanical
system may be time-varying, piecewise linear fits to the phase and the natural log of the
amplitude can be used to identify the system as it evolves with time. Variations on these
concepts in Section 3.7 have been utilized by Ruzzene et al. (1997) and Staszewski
(1997) to illustrate their applicability to linear system identification. Staszewski (1998)
later demonstrated the applicability of the approach for non-linear systems, where the
time-varying estimates of frequency and damping are truly required. Though
Staszewski’s work (1997, 1998) was primarily directed toward mechanical systems with
fundamental frequencies larger than most found in Civil Engineering, Ruzzene et al.
(1997) and Hans et al. (2000) provided examples geared specifically toward Civil
Engineering, applying the techniques to full-scale data, though without any consideration
of the processing concerns detailed in this chapter. When considering Civil Engineering
structures of even longer period, for which the characteristics of narrowbanded response
become increasingly prevalent, these previously unaddressed processing concerns
highlighted in this chapter begin to surface more critically. As these systems are usually
of longer period, frequency resolutions must be refined to insure modal separation. This
371
in turn results in an increase of end effects errors that can have a dramatic influence on
the quality of wavelet amplitudes and modal properties such as damping. The occurrence
of these phenomena is rationalized in this study for the popular Morlet wavelet and a
framework for the analysis of ambient vibration data using wavelets and the Random
Decrement Technique discussed in Chapter 2 is introduced. Demonstrations on simulated
and measured full-scale data within this chapter highlight these important considerations
and the manner in which the provisions in Chapter 4 are applied in wavelet-based system
identification schemes for Civil Engineering applications.
8.3 Influence of End Effects in System Identification: Spectral Measures
One of the more notable signal processing concerns in the identification framework for
wavelets is end effects, introduced previously in Section 4.2.2. Their impacts can be
examined in both the temporal and spectral representations of wavelet transforms. While
the influence of end effects in the frequency domain was clearly visualized in Figure 4.8,
they may be further quantified via time-varying spectral measures. The signal under
consideration, shown in Figure 8.1a, is the free vibration response of a SDOF oscillator
with natural frequency fn of 0.15 Hz and critical damping ratio ξ of 0.01. In the case of a
fo = 1 Hz wavelet analysis, the end effects have no influence on the estimate of
instantaneous frequency from the ridge of the transform, but have a considerable effect
on the amplitude of the wavelet skeleton for the first and last few cycles of oscillation, as
shown in Figure 8.1b,e. The vertical dotted lines mark integer multiples of ∆t for this
analysis in the end effects measure in Equation 4.9. It is not until 3∆t that the end effects
on the wavelet skeleton amplitude diminish. This is rectified by applying the padding
372
x(t)
(a)
(b)
∆x(t), x(t)
x(t)
x(t)
(f)
∆x(t), x(t)
(g)
x(t)
x(t)
(e)
HPBW [Hz]
(d)
(c)
without padding
(h)
without padding
with padding
with padding
t [s]
FIGURE 8.1. (a) Signal; (b) & (e) signal (dark) and wavelet skeleton (light) at
ends without padding; (c) & (f) signal (dark) and wavelet skeleton (light) at ends
with padding; (d) & (g) signal (dark) and difference between signal and wavelet
skeleton (light) at ends with padding; (e) bandwidth estimate with & without
padding
373
operation for β = 4, as shown in Figure 8.1c,f. After doing so, the wavelet skeleton can
hardly be discerned from the actual signal, though as Figure 8.1d,g reveals, a slight
deviation of the amplitude is still present at the very beginning and end of the signal,
though it is small relative to the signal’s amplitude at these points in time. The half-power
bandwidth identified from the wavelet instantaneous spectra is constant, as expected for
this linear oscillator, with the exception of the end effects region. As shown in Figure
8.1h, the bandwidth measure, being more sensitive, is significantly influenced by the end
effects. When padded with β = 4, the bandwidth accuracy is vastly improved, though
Figure 8.1h demonstrates that within the first 3∆t, the bandwidth is still deviating, a result
that cannot be fully improved with larger values of β. This is due to the fact that the
remaining slight inaccuracies in the amplitude lead to a more marked inaccuracy in the
sensitive bandwidth measure. Note also that the bandwidth of the resulting wavelet
instantaneous spectra are larger than their Fourier equivalent, as a result of the
windowing applied by the Gaussian function of the Morlet wavelet, a feature
demonstrated previously Chapter 4.
8.4 Influence of End Effects in System Identification: Time Domain
Time domain system identification on the system in Figure 8.1a may proceed as
discussed in Section 3.7.4. As a result of the reflective padding operation, the estimation
of damping in the time domain is also enhanced, though not completely rectified. As
shown in Figure 8.1e, the deviations in the amplitude with padding are slight and
diminish with each cycle of oscillation, but still have marked impact for the more
sensitive estimation of damping in this system. Note that the bandwidth in the previous
374
section and decay envelope in the previous section are both the major descriptors of
damping in mechanical oscillators and are most significantly impacted by residual end
effects. The error in the wavelet skeleton’s amplitude is most significant at t = 0, though
reduced from 50% to about 5% with the addition of padding. This does not affect the
estimation of instantaneous frequency, which is consistently within 1%. Table 8.1 lists
the identified damping from each cycle of the decay. The time span of each cycle is
provided in parenthesis and less reliable values of damping are shaded in gray. As shown
in column one, without padding, the damping can only be reliably estimated beyond
about 5∆t. Therefore, direct application of the techniques discussed in Ruzzene et al.
(1997) and Staszewski (1998) for such narrowband systems should proceed using only
the wavelet data beyond 5∆t, though the authors make no mention of this nor account for
this in their estimations. However, the padding operation introduced in this study leads to
a vast improvement in the estimates from the first three cycles and produces highly
accurate estimates after only 3∆t, allowing more of the signal to be used in system
identification.
TABLE 8.1
ESTIMATE OF DAMPING FROM WAVELET SKELETON OF SDOF SYSTEM
fo = 1 Hz,
β=0
Critical End Effects 5∆t = 23.57 s
Region
st
1 Cycle (0-6.7 s)
-0.0678
nd
2 Cycle (6.7-13.3 s)
-0.0123
3rd Cycle (13.3-20.0 s)
0.0067
th
4 Cycle (20.0-26.7 s)
0.0099
fo = 1 Hz,
β=4
3∆t = 14.14 s
fo = 0.5 Hz,
β=4
3∆t = 7.07 s
fo = 0.25 Hz,
β=4
3∆t = 3.53 s
0.0034
0.0084
0.0098
0.0099
0.0054
0.0099
0.0099
0.0099
0.0097
0.0102
0.0101
0.0101
375
Note that since the identification of this system is proceeding in the time domain,
a reduced value of fo can be used to diminish end effects, though sacrificing the frequency
resolution. When doing so, it becomes possible to identify the damping within 1%
accuracy using virtually the full length of padded data, as demonstrated in the last two
columns of Table 8.1. While this is attractive, it is demonstrated in the following section
that this may not be plausible for MDOF systems, particularly those with closely spaced
modes.
8.5 System Identification from Free Vibration: MDOF Example
Additional issues associated with wavelet system identification are can be demonstrated
through a MDOF example, where the issues of modal separation can be most clearly
investigated. Recall that the Morlet wavelet-based analysis allows flexibility in the value
of the central frequency fo to obtain desired resolutions, as discussed previously in
Chapter 4. This selection becomes critical if closely spaced modes are suspected.
Staszewski (1997) discussed the use of shifted Morlet wavelets for separation of closely
spaced, high frequency modes; however, in the case of low-frequency systems, the
judicious selection of the central frequency of the Morlet wavelet can similarly
accomplish the same operation directly. This is illustrated by the analysis of 100 seconds
of data (shown in Figure 8.2), sampled at 10 Hz from the impulse response of the
following system MDOF system
m&x& + cx& + kx = δ(t )
376
(8.1)
IRF
t [s]
FIGURE 8.2. Impulse response function of MDOF system
with closely spaced modes
where m, c and k are the mass, damping and stiffness matrices, respectively, x is the
vector of displacements, x& and &x& are the first and second derivatives of x, and δ (t) is
the unit impulse function. In order to achieve the desired frequency characteristics, the
following mass and stiffness matrices were defined
⎡ 4 −1 0 ⎤
m = mI and k = k ⎢⎢− 1 2 − 1⎥⎥
⎢⎣ 0 − 1 4 ⎥⎦
(8.2 a,b)
where I is the identity matrix, m=1000 kg and k=10 kN/m. As the system is assumed to
have a critical damping ratio ξ = 0.01 in each mode, a damping matrix can be defined as
[
c = m ΦM
−1
⎡2ξω1 M 1
⎢ 0
⎢
⎢⎣ 0
]
0
2ξω 2 M 2
0
⎤
⎥ M −1Φ T m
0
⎥
2ξω 3 M 3 ⎥⎦
0
[
]
(8.3)
where M1,2,3 are the modal masses of the modal mass matrix M, the modal frequencies are
ω1,2,3 = 2 π f1,2,3, and Φ is the matrix of mode shapes. The response at the third degree of
freedom of the MDOF system as the result of a unit impulse at that location was then
377
generated via state-space simulation of Equation 8.1. Note that the stiffness and mass
matrices in Equation 8.2a,b were selected to achieve a response with both low frequency
content and closely spaced modes. The resulting frequencies are 0.567, 1.006 and 1.095
Hz, and all three modes have a critical damping ratio of 0.01. The latter two modes are
within 10% of one another, requiring a refined frequency resolution, and thus providing
an ideal venue in which to explore the significance of fo for optimal modal separation.
Without a priori knowledge of the system, the selection of a central frequency for
analysis should initiate from information gathered through a visual inspection of the time
series, as discussed in Chapter 5. Such inspection indicates that one obvious period of
oscillation is on the order of 1 s. As a general rule of thumb, a frequency resolution of
one-tenth the period of oscillation is desirable, i.e. ∆f1,2 = 0.1 Hz. This serves as a starting
point for the analysis and may be refined even further in subsequent analyses to uncover
additional details. For the purposes of demonstration, varying levels of α in Equation 4.6
are used to demonstrate the influence of analysis window overlap in the adjacent
frequency bands. Assuming α = 1 defaults to a direct application of Gabor’s mean square
duration and necessitates a minimum wavelet central frequency of 2.25 Hz. To simplify,
3 Hz is chosen for analysis. As suggested previously, α = 2 and 3 provide a more
accurate means of modal separation. To demonstrate the influence of these parameters, fo
= 6 Hz and 8 Hz wavelet analyses are also performed on the same signal. Table 8.2 lists
the resulting frequency and temporal resolutions as defined by Equations 4.5 and 4.8.
378
TABLE 8.2
WAVELET-BASED IDENTIFICATION OF MDOF SYSTEM WITH CLOSELY
SPACED MODES
M
O
D
E
Resolutions
∆ti
∆fi
[s]
[Hz]
fn
[Hz]
Actual
ξ
1
2
3
3.74
2.11
1.94
0.021
0.038
0.041
0.567
1.006
1.095
0.01
0.01
0.01
1
2
3
7.49
4.21
3.87
0.011
0.019
0.020
0.567
1.006
1.095
0.01
0.01
0.01
1
2
3
9.98
5.62
5.17
0.008
0.014
0.015
0.567
1.006
1.095
0.01
0.01
0.01
1
2
3
7.49
4.21
3.87
0.011
0.019
0.020
0.567
1.006
1.095
0.01
0.01
0.01
1
2
3
9.98
5.62
5.17
0.008
0.014
0.015
0.567
1.006
1.095
0.01
0.01
0.01
Analytic Signal ID
avg[fn]
avg[ξ] CoV[ξ]
[Hz]
Ridge IF & Log Dec ID
avg[fn]
avg[ξ] CoV[ξ]
[Hz]
fo=3 Hz, β=0
Incomplete
Modal Separation
fo=6 Hz, β=0
0.567
0.0098
1.006
0.0099
1.095
0.0098
fo=8 Hz, β=0
0.567
0.0098
1.006
0.0098
1.095
0.0098
fo=6 Hz, β=4
0.567
0.0099
1.006
0.0100
1.095
0.0100
fo=8Hz, β=4
0.567
0.0099
1.006
0.0100
1.095
0.0100
4.48%
4.43%
10.0%
0.566
1.005
1.094
0.0099
0.0099
0.0098
4.61%
8.15%
10.4%
6.58%
4.79%
3.98%
0.565
1.008
1.098
0.0097
0.0097
0.0096
5.93%
9.48%
12.5%
1.12%
1.53%
8.93%
0.566
1.005
1.095
0.0100
0.0101
0.0100
2.47%
7.09%
13%
1.74%
1.13%
0.91%
0.565
1.008
1.098
0.0099
0.0101
0.0098
2.91%
8.58%
10.4%
8.5.1 Ridge Extraction and Wavelet Instantaneous Spectra
The MDOF system identification procedure employed here is schematically represented
in Figure 8.3. In this case the input to the wavelet framework comes directly from
impulse response functions, following Track B in the figure. As shown in Figure 8.4a, the
wavelet instantaneous frequencies identified from the ridges for fo = 3 Hz successfully
identify the first mode, but have difficulty fully separating the second and third modes,
clearly indicating that the choice of central frequency did not provide adequate frequency
resolution. The ridge identification using Equation 4.25 becomes very poor beyond 80 s –
379
Measured Response
Random Decrement
Signature
Measured Free
Vibration/Impulse
Response
t [s]
t [s]
t [s]
ln[A(t)]
φ(t)
ωn
or
-ξωn
A
B
CWT
t [s]
Re[W(a,t)]
Re[W(a=ar,t)]
t [s]
t[s]
f [Hz]
t [s]
Wavelet Scalogram
Wavelet Skeleton
Amplitude & Phase
Random
Decrement
Technique
Extract Skeleton
FIGURE 8.3. Wavelet system identification framework, Track A for ambient
vibrations or Track B for measured free vibration or impulse response
380
f [Hz]
(b
(c)
(d
f [Hz]
(a)
t [s]
t [s]
FIGURE 8.4. Instantaneous frequencies identified from ridges of wavelet transform
for (a) fo = 3 Hz, (b) fo = 6 Hz, (c) fo = 8 Hz and (d) fo = 6 Hz with padding
a direct consequence of the diminished amount of signal energy at this point and
demonstration of the influence of end effects. Figure 8.4b reveals that separation was
possible for fo = 6 Hz, again with some difficulty at the final seconds of the signal. The
analysis for fo = 8 Hz in Figure 8.4c merely extends the end effect region deeper into the
signal. With the introduction of the padding scheme introduced in Chapter 4 to the case
of fo = 6 Hz, the slight end effects at the initiation of the signal are remedied in Figure
8.4d and diminished at the termination of the signal, although the lack of signal energy in
this region makes identification of the highest mode difficult.
In general, the influence of end effects is not observed to significantly influence
instantaneous frequencies, as discussed in Section 8.4. To demonstrate, the instantaneous
spectra are provided in Figure 8.5, verifying the concentration of energy at the ridge
frequencies. Note that Figures 8.5a and 8.5b illustrate the intermittency of modal
separation evident in the first mode ridge (Figure 8.4a) for the fo = 3 Hz analysis. Further,
the progressive narrowing of the instantaneous spectral bandwidth within the frames,
381
(b)
(c)
(d)
|W(a,t=tj)|2
|W(a,t=tj)|2
(a)
f [Hz]
f [Hz]
FIGURE 8.5. Instantaneous wavelet spectra for
(a) fo = 3 Hz when two modes are present, (b) fo
= 3 Hz when three modes are present, (c) fo = 6
Hz and (d) fo = 8 Hz
thereby indicating more complete modal separation, illustrates the refinement of
frequency resolution as central frequency is increased.
8.5.2 Wavelet Skeletons and End Effects
The wavelet skeleton introduced in Chapter 3 can be extracted from the ridges identified
in Figure 8.3, with the real component being proportional to the signal itself. Figure 8.5a
reiterates the inability of the fo = 3 Hz analysis to separate the two higher modes. The
analyses in Figures 8.5b and 8.5c further demonstrate that such modal separation is
possible with sufficient frequency resolution. It was noted in Section 8.4 that, in general,
the wavelet skeletons are not capable of accurately capturing the amplitude of the signal
382
for the first 3∆ti due to the end effects phenomenon, requiring β = 3 in Equation 4.9.
These critical regions are marked by the vertical dotted line in each plot. Within this
region, the initiation of the impulse response function in Figure 8.6 manifests a rounded
hump, very evident in the second mode in Figures 8.6b and 8.6c. In addition, at the end of
the signal, a flare in amplitude also occurs in this 3∆ti region, intensifying with time.
Particularly in the case of fo = 8 Hz, the end effects regions for the first mode can
consume a significant portion of the signal. It becomes evident that for high-resolution
analyses, the end effects at low frequencies leave little useable signal for reliable system
identification. Note that such marked end effects were not apparent in previous work due
to the smaller central frequencies employed, as a result of the attention toward higher
frequency phenomenon and the lack of closely spaced modes. However, in the analysis of
many Civil Engineering structures, such manifestations should be expected. In efforts to
diminish the presence of end effects, the reflective padding procedure of Chapter 4 is
employed in Figure 8.6d, revealing the marked improvement in the wavelet
approximations of the signal, shifting the vertical bars denoting the 3∆ti regions engulfed
by end effects to t = 0 and t = T = 100 s, essentially preserving the entire signal.
8.5.3 System Identification via Wavelet Amplitude and Phase
Figure 8.7 displays the phase and amplitude curves of the wavelet-transformed data for
each mode, later used to identify frequency and damping for the fo = 6 and 8 Hz analyses,
which produced meaningful wavelet skeletons. For this system with constant dynamic
properties, these should be straight lines, though some minor rippling occurs in the
amplitude envelopes, particularly near the end effects regions. Using the analytic signal
383
Re[W(ar,t)]
(a)
(b)
(c)
(d)
t [s]
FIGURE 8.6. Real component of wavelet skeleton – each panel
contains skeleton for the ridge associated with modes 1, 2 and 3,
respsectivley, for (a) fo = 3 Hz, (b) fo = 6 Hz, (c) fo = 8 Hz and (d)
fo = 6 Hz with padding. Dotted vertical line demarks the 3∆ti
region of end effects
(b)
ln[A(t)]
φ(t)
(a)
ωD
ωD
−ξωn
−ξωn
ωD
ωD
−ξωn
−ξωn
ωD
ωD
−ξωn
−ξωn
t [s]
FIGURE 8.7. Wavelet phase and amplitude curves for system identification of
MDOF system (solid) with linear least squares fit (dotted), for (a) fo = 6 Hz and (b)
fo = 8 Hz. Each panel contains data for modes 1, 2 and 3 from left to right
384
theory discussed previously, the frequency and damping may be identified via Equations
3.35 and 3.34, through a piecewise linear fit to the phase and the natural log of the
wavelet amplitude along the ridges, respectively. While the identification of frequency is
generally not difficult, damping proves to be a far more elusive parameter to identify. As
shown in Figure 8.8, the piecewise fits of the amplitude curves in Figure 8.7 produce
damping estimates that gradually approach the actual damping of 0.01. The initial
inaccuracies are the result of end effects, producing negative values of damping in
Figures 8.8a and 8.8c due to the rounding of the skeleton shown in Figures 8.6b and 8.6c.
As discussed in Section 8.4, without padding, damping values do not stabilize until 5∆ti,
marked by the third vertical bar in each plot. By introducing the padding operation,
Figures 8.8b and 8.8d, the signatures do not manifest negative damping and stabilize
within 3∆ti. As discussed previously, the manifestation of end effects in the wavelet
amplitudes can be minimized through the reflective padding procedure; however, slight
inaccuracies in the amplitude remain, on the order of a few percent, for the first 3∆ti. For
the more sensitive bandwidth measures, the presence of even slight errors in the
amplitude translates into more significant deviations in parameters like damping. Still,
the introduction of padding eliminates negative damping estimates and stabilizes the
damping estimate sooner, though system identification for the purpose of damping
estimation should not be performed on the first or last 3∆ti of the wavelet skeleton. In
light of this, the fact that many Civil Engineering structures possess very low levels of
damping is actually a benefit, as the IRFs will take longer to decay, leaving adequate
amounts of data for analysis despite neglecting some regions.
385
(b)
(c)
(d)
ξ
(a)
t [s]
FIGURE 8.8. Identification of damping from piecewise fit to amplitude curves
of MDOF system for (a) fo = 6 Hz, (b) fo = 6 Hz with padding, (c) fo = 8 Hz,
and (d) fo = 8 Hz with padding. Each panel contains data for modes 1, 2 and 3
from left to right. Dotted vertical line demarks the 3∆ti, 4∆ti and 5∆ti end
effects regions
By constraining the identification of the system to the regions beyond 3∆ti of
ends, the frequency and damping can be identified on average with considerable
accuracy, as summarized in Table 8.2 provided previously in Section 8.5. The natural
frequencies of the system show little fluctuation and are identified with precision and
manifest little sensitivity to end effects. The damping estimates, in terms of their mean
and coefficient of variation (CoV), are also quite reliable, though they temporally vary.
The introduction of padding to the systems, results in a decrease in the CoV, again
affirming the efficacy of this pre-processing tool. The increase in central frequency does
not markedly affect the statistics of the damping estimation, though Figure 8.8
demonstrates that the behavior temporally is more reasonable. Therefore, if estimates of
frequency and damping are to be obtained for a known linear system, a linear fit to the
entire curve in Figure 8.7 may be performed to yield estimates similar to the averaged
quantities in Table 8.2, and a value of α = 2 should be sufficient. However, in the case of
386
nonlinear systems, as explored by Staszewski (1998), a piecewise analysis is necessary to
capture time-varying dynamic properties. However, note the rippling of damping
estimates in the fo = 6 Hz analysis (Figures 8.8a and 8.8b). The rippling in this case is
indicative of incomplete modal separation. While the frequency ridges in Figure 8.4b
indicated that separation was successful, it is again the more sensitive damping measure
that verifies that the bandwidths of the two systems are still slightly overlapping. This is
evident when comparing Figures 8.5c and 8.5d. The increase in fo eliminates this spurious
behavior (Figures 8.8c and 8.8d), though again at the expense of temporal resolution,
requiring some compromise between achieving modal separation and minimizing end
effects. Unfortunately, a choice of too small a fo may produce rippling such as that in
Figure 8.8a that may be mistaken for nonlinear behavior. Therefore, if the assumption of
a linear system cannot be safely made, α = 3 may be more appropriate for time-varying
system identification. Clearly for low frequency Civil Engineering systems this can lead
to a significant loss of data.
For comparison, the frequencies corresponding to the scales of the ridges are
listed in Table 8.2 and are reasonably accurate estimates of the natural frequency. As a
result, for time-frequency analysis such as that presented in Sections 8.3 and 8.4,
frequencies can usually be identified solely from the amplitude of the wavelet transform
for α = 1 or α = 2 if closely spaced modal components are suspected. For further
comparison, an alternative means for damping estimation is provided by a logarithmic
decrement approach applied to wavelet skeletons (Hans et al., 2000; Lamarque et al.,
2000). Such an approach, being reliant on the peaks of the amplitude decay, is more
susceptible to fluctuations, particularly in the higher modes, even when modal separation
387
ξ
t [s]
FIGURE 8.9. Identification of damping via logarithmic decrement of
wavelet skeleton for fo = 8 Hz with padding; modes 1, 2 and 3 shown
from left to right
is achieved. Figure 8.9 demonstrates this for the logarithmic decrement identification of
damping for fo = 8 Hz with padding applied. Note the stabilization trend in the damping
witnessed in Figure 8.8 is again apparent, though with more irregular variations. The
statistics of the logarithmic decrement identification are also provided in Table 8.2 for
comparison and reveal that the damping estimates are reasonable in the mean. Without
padding, the CoV of the logarithmic decrement results is comparable to those obtained
using the method based on analytic signal theory. However, when padding is applied and
complete modal separation is assured (fo = 8 Hz, β = 4), the CoV of the logarithmic
decrement technique is significantly larger than that derived from the analytic signal
identification approach presented in Figure 8.3, especially for the higher modes. This
highlights that much of the variance in damping identified by the analytic signal approach
is merely due to end effects and the lack of modal separation.
388
8.6 System Identification from Ambient Vibration: MDOF Example
Before the framework denoted as Track A in Figure 8.3 is introduced for application to
actual measured data, the efficacy of the approach must be assessed. This assessment is
conducted utilizing a simulated two-degree-of-freedom (2DOF) system, with frequencies
of 0.2757 Hz and 0.3559 Hz and critical damping ratios of 0.01. The system is repeatedly
simulated, progressively increasing the length of simulated data: Case 1 is comprised of 5
hours of data sampled at 10 Hz, Case 2 extends this to 10 hours and Case 3 to 30 hours.
Intuitively, one would expect that the decrement signature more closely approximates the
true impulse response or decay signature of the 2DOF system as the number of averages
in the decrement increases. This generally holds true from repeated simulations of Cases
1, 2 and 3. However, it is observed that the agreement between the RDS and the signal
decay diminishes in the transition zones between pockets of higher amplitude response, a
consequence of the two modes beating, as shown in Figure 8.10 in the comparison of
RDS obtained using the strictest RDT with local extrema trigger condition (see Chapter
2) with the actual impulse response function for the system. As a result, it can take a
considerable number of averages to produce decrement signatures that more closely
approximate this IRF, dependent on the random characteristics of the response within that
simulation. Note in Figure 8.10 that the ability to replicate these low amplitude
transitional ranges is limited. Though the periodicity is well captured, deviations in the
amplitudes, even in the high amplitude peaks, are a clear sign of the difficulty in
obtaining realistic envelope curves via RDT, and thereby accurate estimates of damping.
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Case 1
Case 2
Time [s]
Time [s]
Case 3
Time [s]
FIGURE 8.10. Random decrement signatures (solid) in comparison to actual
impulse response function (dashed) for 2DOF system
The system identification methodology overviewed as Track A in Figure 8.3 is
applied to the Random Decrement Signatures, with fo = 2 Hz, β = 4 and OF = 1. In order
to isolate the errors associated with inaccuracies in the Random Decrement Signature
from those associated with the modal decoupling and system identification using the
wavelet approximation to the analytic signal, the frequencies and dampings estimated
from the RDS using this wavelet technique (Track A) are compared to those estimated
from the actual system IRFs (Track B) using the same wavelet technique. The results for
all three cases are chronicled in Table 8.3.
Table 8.3 demonstrates that the problem is not with the analytic signal
identification technique or modal separation using wavelets, as all Track B identifications
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using the IRF directly are within a few percent of the actual values. It appears instead to
be tied to the RDT itself. The use of random decrement technique to obtain the decay
signatures can have some uncertainty in general, as discussed in Section 2.3, and even
more pronounced in the case of systems with modes that have close proximity where
some beating of the envelope function occurs. In some cases the RDT may have
difficulty in fully removing the random component of the response. For example, in Case
1, with only limited data, reveals good performance for Mode 1 but not for Mode 2.
TABLE 8.3
WAVELET FREQUENCY AND DAMPING ESTIMATES FROM RDS (TRACK A)
AND IRF (TRACK B)
Case 1
Case 2
Case 3
Track A
f1
ξ1
[Hz]
0.2764 0.0096
0.2753 0.0088
0.2755 0.0089
Track B
f1
[Hz]
0.2753
0.2753
0.2757
Track A
ξ1
0.0097
0.0097
0.0100
f2
[Hz]
0.3561
0.3558
0.3560
Track B
ξ2
0.0127
0.0105
0.0086
f2
[Hz]
0.3559
0.3559
0.3559
ξ2
0.0102
0.0102
0.0102
Doubling the amount of random data, the estimate of Mode 2 is enhanced but the
estimate of Mode 1 degrades to some extent. The use of thirty hours of data in Case 3
produced underestimates of damping that are comparable if not reduced in quality
compared to the data from Case 2. In fact, Case 2 gives the best overall estimate of
damping in both modes, the reason for which becomes very clear when looking at how
well the RDS approximates the IRF in Figure 8.10. The fact that Cases 1 and 3 produced
less accurate damping estimates is clearly evident from Figure 8.10, demonstrating their
inability to capture decay amplitudes and transition regions well. Thus it would seem that
391
Damping Ratio
Frequency [Hz]
(a)
Damping Ratio
Frequency [Hz]
(b)
Simulation Number
Simulation Number
FIGURE 8.11. Frequency and damping identification by RDT and wavelet transform
framework for repeated simulations of (a) Case 1 and (b) Case 2
the presence of more data does not offer clear improvement in the ability to accurately
estimate damping. Moreover, this trend merely echoes the findings of Section 2.3, where
the inherent randomness associated with a given simulation was found to influence the
quality of the RDS and the ensuing damping identification.
To demonstrate this is again the case for this 2DOF system, Cases 1 and 2 were
simulated several times and the Track B approach was applied to each time history. The
results are shown in Figure 8.11. As expected, the variability in frequency estimation is
limited, however, in terms of the identified damping levels, the variation between
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simulations reflects the themes observed in Chapter 2. The level of variability is
somewhat reduced when more data is considered, but again highlights the influence that
the randomness of the process has on the performance of RDT. From the simulations of
Case 2, with the aid of greater lengths of data, the average damping ratio in Mode 1 is
conservatively estimated as 0.0085 and in Mode 2 as 0.0096. The good performance
noted in Simulation number 2 for Case 2 (Fig. 8.11b) is the result of an RDT signature
that closely approximated the actual IRF due to the random components of the response
being fully cancelled in the averaging. The fact that this is not always achieved further
emphasizes the need for accompanying measures such as the bootstrap approach in the
Appendix of this study to evaluate the level of variance in decrement signatures.
Examples of some of the intermediate steps in this identification procedure are
shown in Figure 8.12. Obviously, the breakdowns of the RDS due to increasing variance
in the signature limits the amount of useable data. The progressive difficulty in ridge and
skeleton extraction is evident in Figure 8.12a. Figure 8.12 demonstrates that the phase
relations are about equal in the identification by both tracks and the damping in the first
mode shows similar features. However, it is the higher mode damping that presents more
obvious challenges, as the degree of rippling is far more pronounced in Track A again
due to limitations in the RDT in recreating the proper IRF.
These examples and others within this chapter emphasize the interplay between
the various parameters in the wavelet framework in Chapter 4 and the competing interests
of the decrement signatures discussed in Chapter 2, particularly with respect to the end
effects phenomenon, which, as shown in the beginning of this chapter, still has residual
393
Skel. 2
Skel. 2
Skel. 1
(b) Track B
Skel. 1
(a) Track A
Time [s]
Mode 1
Time [s]
Mode 1
ln [A(t)]
Mode 2
Mode 2
φ(t)
φ(t)
Mode 2
ln [A(t)]
Mode 1
ln [A(t)]
ln [A(t)]
φ(t)
Mode 1
φ(t)
Time [s]
Mode 2
Time [s]
FIGURE 8.12. Wavelet-based system identification from (a) Random Decrement
Signatures and (b) Impulse Response Functions
effects on damping estimation, even when reflective padding is utilized. As a result, the
identification from both the IRF and RDS within this section is performed only on the
useable signal components, lying beyond 3∆ti of the padded signal. Within this region,
the IRF is understandably the more consistent performer. However, to the benefit of
applications to ambient vibrations, the prevalence of the first mode in the response
usually permits lower values of central frequency than those required for systems such as
those in Section 8.5, in which the responses are characterized by two closely spaced
modes comparably participating to the response. As will be shown in the subsequent
example, the dominant first mode response under wind does not typically result in
394
significant excitation of higher modes and thus this technique can be quite attractive for
application, as the higher modes and noise make minor contributions and need only to be
low pass filtered from the response, allowing lower central frequency values and thereby
reduced end effects regions.
8.7 System Identification from Ambient Vibration Data: Full-Scale
Example
Obtaining the IRF or free vibration curve can be difficult if not impossible for many Civil
Engineering structures, as controlled testing can be costly in terms of equipment or
disruptive to the daily function of the building. As a result, there is interest in developing
approaches to permit the extraction of decay curves from ambient vibration data — an
oftentimes challenging proposition. The Random Decrement Technique has evolved as a
popular analysis tool, as discussed in greater detail in Chapter 2. Though Ruzzene et al.
(1997) employed this technique to analyze full-scale bridge data, some added concerns
surface when the wavelet system identification approach discussed here is merged with
the RDT for low frequency systems. Chapter 2 has shown that the variance of RDS
signatures increases with each cycle of oscillation as one moves further from the trigger
condition, indicating that system identification should be restricted to the first few cycles
of the RDS. A variance envelope on the decrement signature, developed in Chapter 2 and
the Appendix of this study, illustrates, via Figure 8.13, the limited number of cycles over
which identification can be reasonably made. The degradation of decrement signatures is
problematic for low-frequency Civil Engineering structures, when wavelets are employed
in the framework of Figure 8.13 in Track A, since the end-effect region described in
395
RDS
t [s]
FIGURE 8.13. Example of Random
Decrement Signature (gray) and variance
envelope
Equation 4.9 lengthens. While padding does repair the amplitudes considerably, reliable
damping identification should only proceed outside of the 3∆ti end effects region, though
frequency identification and signal amplitude reconstruction remains viable throughout.
The amount of data lost can be offset to some extent by decreasing the analyzing
frequency fo of the Morlet wavelet, albeit compromising the ability to distinguish closely
spaced modes. This permits the identification of damping from more reliable portions of
the RDS. As motivated above, this compromise can be justified for dominant first mode
responses under wind.
To illustrate, the RDT is applied to 1.5 hours of measured full-scale acceleration
data, sampled at 20 Hz, a portion of which is shown in Figure 8.14. The data was
measured along the y-axis of a tower in Japan during a typhoon (Tamura et al., 1993).
Note that during simple free vibration tests, the tower was found to have a fundamental
sway period of 1.6 seconds in both directions, with a critical damping ratio of 0.015. The
396
ay [milli-g]
t [s]
FIGURE 8.14. Ten-minute segment of the acceleration
response (y-dir) of tower in typhoon
application of RDT in the strict peak trigger condition yielding 225 averaged segments,
yields the signature shown in Figure 8.15a. Note the signature lacks the smooth
characteristic one would anticipate from a decay of a SDOF damped oscillator, indicating
the presence of higher modes or other noise in the RDS, which would normally have to
be separated through some bandpass filtering. However there is no characteristic beating
in the RDS, observed previously in the presence of closely spaced modes in Section 8.6,
allowing the relaxation of the wavelet resolutions. Therefore, a very relaxed resolution
wavelet (fo=0.5 Hz) may be applied. The low value of central frequency then allows the
maximization of the amount of usable transformed signal in light of the competing
restrictions of variance in the Random Decrement Signature and the end effects of the
wavelet transform, minimized to some extent with the padding operation. The real
component of the wavelet coefficients, given in Figure 8.15b, identifies a single mode
contributing to the response, as typically observed under wind excitation, and the
decaying oscillatory character of the decrement signature. The breadth of the scalogram
in the frequency domain reiterates the loss in frequency resolution, which resulted from
the choice of a Morlet wavelet with superior time resolution. The skeleton extracted from
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(a)
Re[W(a,t)]
RDS
(b)
f [Hz]
Re[W(a=ar,t)]
t [s]
(c)
Re[W(a=ar,t)]
t [s]
(d)
t [s]
FIGURE 8.15. (a) Random decrement signature, (b) real component of wavelettransformed Random Decrement Signature in 3D; (c) real-valued skeleton; (d) zoom
of real-valued skeleton (solid) with theoretical skeleton (dotted) for fn = 0.625 Hz and
ξ = 0.015
the ridge of the wavelet is shown in Figure 8.15c and is clean and smooth, as the wavelet
transform has separated the higher frequency noise in the RDS in Figure 8.15a. The RDS
embodied by the skeleton appears to be relatively stable up to about 10 seconds, after
which it degrades in quality as a result of the increasing variance shown in Figure 8.13.
As a result, the system identification should be performed at most on the first 10 seconds
of the RDS. Figure 8.15d shows a comparison between the signature in the first 10
398
seconds and the anticipated signature, based on the structure’s known frequency and
damping. Inconsistencies in amplitude and phase between the two propagate with time, as
a result of RDS variance, and suggest that system identification should be performed
within the first 3-4 seconds of the RDS. Though the comparison in Figure 8.15d is not
available during the typical system identification process, the variance envelopes shown
in Figure 8.13 allow the user to make a similar conclusion that the RDS quality is suspect
beyond 4 seconds.
As discussed previously, due to minor end effects remaining in spite of padding,
identification of damping should proceed beyond 3∆t1 of the initiation of the wavelettransformed Random Decrement Signature. Thus the useable portion of the RDS may be
first defined as, tuse = 1.66 to 10 seconds. The end effects region was minimized as a
result of compromising frequency resolution, which will not significantly affect the
results for this system since no closely spaced modes are present. Identification of
frequency and damping by the procedure based on analytic signal theory produced
estimates of frequency and damping shown in Figures 8.16a and 8.16c. Note that beyond
10 seconds, the quality of estimates rapidly degrades, due to the variance of the RDS.
Zooming in on the first 10 s in Figures 8.16b and 8.16d reaffirms two previous
observations: the identified frequency suffers very little as the result of end effects, and
even with the padding operation, damping estimates are less reliable in the first 3∆t1
(marked by the dashed line in Figure 8.16d) though gradually approaching more stable
levels. As summarized in Table 8.4, piecewise fits of the amplitude and phase of the RDS
over only tuse produced mean frequency and damping estimates consistent with that
observed from free vibration testing, though with considerable variance in the damping
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(a)
(b)
φ(t)
fn [Hz]
(e)
(f)
(d)
ξ
ln[A(t)]
(c)
ωn
-ξωn
t [s]
t [s]
FIGURE 8.16. (a) Frequency and (c) damping identified from wavelet skeleton,
zoom of (b) frequency and (d) damping estimates, (e) wavelet phase along ridge,
(f) wavelet amplitude along ridge
estimate. Note that the tracking of nonlinear frequency and damping characteristics via
the Random Decrement Technique, discussed by Tamura & Suganuma (1996), can be
accomplished by varying the trigger condition on captured segments. The resulting
decrement signature will manifest, in its first few reliable cycles, the frequency and
damping referenced to the amplitude level defined by the trigger. Therefore, the RDS
associated with a given trigger level can be assumed linear and any nonlinearity will be
evident as the trigger level is changed and other RDS are analyzed.
In such cases where the RDS is known to represent a linear system, it is more
reasonable to conduct a best fit of the entire length of tuse as shown in Figures 8.16e and
8.16f. Note that there is still a slight deviation in the log of the amplitude, producing a
slightly smaller damping estimate than the piecewise mean. However, re-inspection of
400
Figure 8.15d reveals that significant deviations in phase and amplitude in the decrement
signature are evident beyond the 4th second, a fact affirmed by Figure 8.16. Therefore,
restricting the identification to tuse = 1.657 - 4 s, is more reasonable. The resulting
estimates of damping (0.0151) and frequency (0.645 Hz), are both within a few percent
of the values observed in free vibration test. These results are also consistent with the
findings of Tamura et al. (1993) from data collected during the passage of several
typhoons. The results in Table 8.4 and Figure 8.16 highlight the importance of
identification in the early stages of Random Decrement Signatures. It is interesting to
note that Ruzzene et al. (1997) found some discrepancy between the identified damping
values and those observed previously by other techniques, possibly due to estimation of
damping from the later, less-reliable cycles of the RDS. Such characteristics of the RDT
make it vital that the end effects issues in wavelet transformed Random Decrement
Signatures are recognized and accounted for to insure reliable system identification.
TABLE 8.4
WAVELET-BASED IDENTIFICATION FROM FULL-SCALE AMBIENTLYEXCITED DATA
Measured in Free
Vibration Test
fn
ξ
[Hz]
0.625
0.0150
Identified Piecewise over tuse
fn
[Hz]
0.650
avg[ξ]
CoV[ξ]
0.0141
18.43%
401
Identified from
tuse = 3∆t1 - 10 s
fn
ξ
[Hz]
0.651
0.0136
Identified from
tuse = 3∆t1 - 4 s
fn
ξ
[Hz]
0.645
0.0151
8.8 Conclusions
The last several chapters have demonstrated the utility of wavelet transforms in analyzing
a variety of signals in Civil Engineering, for not only time-frequency analysis but also for
system identification. This popularity stems in part from its multi-resolution capabilities
and the potential for direct relations to the Fourier transforms. This chapter focused on
some of the processing concerns which surface in Civil Engineering applications and
introduced a system identification framework applicable to measured free vibrations or
ambient vibration data.
The discussions and examples in this chapter emphasized that application of the
Morlet wavelet requires judicious selection of central frequency in light of the resulting
time and frequency resolutions, a fact that becomes significant for Civil Engineering
structures, whose dynamics are often more narrowbanded than traditional mechanical
systems. For such systems, the presence of end effects can compromise the accuracy of
wavelet skeletons and have even more marked effects on bandwidth measures. While the
padding operation is demonstrated to improve the scalogram amplitudes thereby
eliminating the appearance of negative damping, and more quickly stabilizing the
behavior of damping estimates, reducing their coefficient of variation. However, the
sensitivity of the damping measure still results in some inaccuracy within 3∆ti of the
ends, defining the acceptable analysis regimes of the transformed signal. Though an
improvement over the results without padding, to minimize this effect, the central
frequency should be kept to the smallest value possible without compromising the ability
to separate closely spaced modes, an important consideration when the Random
402
Decrement Technique is applied for analysis of ambient vibration data. However, it was
observed the when adequate modal separation was achieved and padding was applied, the
coefficient of variation in the analytic signal approach is quite low, highlighting that
much of the variance in damping estimates may be attributed to end effects and modal
overlap and not the identification technique itself. In total, the discussions in Chapter 2
and this chapter highlight the challenges facing Civil Engineers in the identification of
damping from ambient vibration. The next chapters now discuss the manner in which
such ambient vibration data can be obtained.
403